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Properties

Label 2-5e2-25.17-c2-0-2
Degree 22
Conductor 2525
Sign 0.933+0.358i0.933 + 0.358i
Analytic cond. 0.6812000.681200
Root an. cond. 0.8253480.825348
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.259 + 0.508i)2-s + (0.838 − 5.29i)3-s + (2.15 + 2.97i)4-s + (−3.73 + 3.32i)5-s + (2.47 + 1.79i)6-s + (1.66 + 1.66i)7-s + (−4.32 + 0.685i)8-s + (−18.7 − 6.09i)9-s + (−0.724 − 2.76i)10-s + (0.984 + 3.02i)11-s + (17.5 − 8.94i)12-s + (1.52 + 3.00i)13-s + (−1.27 + 0.414i)14-s + (14.4 + 22.5i)15-s + (−3.76 + 11.5i)16-s + (−2.34 − 14.7i)17-s + ⋯
L(s)  = 1  + (−0.129 + 0.254i)2-s + (0.279 − 1.76i)3-s + (0.539 + 0.743i)4-s + (−0.746 + 0.665i)5-s + (0.412 + 0.299i)6-s + (0.237 + 0.237i)7-s + (−0.541 + 0.0857i)8-s + (−2.08 − 0.677i)9-s + (−0.0724 − 0.276i)10-s + (0.0894 + 0.275i)11-s + (1.46 − 0.745i)12-s + (0.117 + 0.230i)13-s + (−0.0911 + 0.0296i)14-s + (0.965 + 1.50i)15-s + (−0.235 + 0.724i)16-s + (−0.137 − 0.869i)17-s + ⋯

Functional equation

Λ(s)=(25s/2 ΓC(s) L(s)=((0.933+0.358i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(25s/2 ΓC(s+1) L(s)=((0.933+0.358i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2525    =    525^{2}
Sign: 0.933+0.358i0.933 + 0.358i
Analytic conductor: 0.6812000.681200
Root analytic conductor: 0.8253480.825348
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ25(17,⋅)\chi_{25} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 25, ( :1), 0.933+0.358i)(2,\ 25,\ (\ :1),\ 0.933 + 0.358i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.914845−0.169521i0.914845 - 0.169521i
L(12)L(\frac12) ≈\approx 0.914845−0.169521i0.914845 - 0.169521i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(3.73−3.32i)T 1 + (3.73 - 3.32i)T
good2 1+(0.259−0.508i)T+(−2.35−3.23i)T2 1 + (0.259 - 0.508i)T + (-2.35 - 3.23i)T^{2}
3 1+(−0.838+5.29i)T+(−8.55−2.78i)T2 1 + (-0.838 + 5.29i)T + (-8.55 - 2.78i)T^{2}
7 1+(−1.66−1.66i)T+49iT2 1 + (-1.66 - 1.66i)T + 49iT^{2}
11 1+(−0.984−3.02i)T+(−97.8+71.1i)T2 1 + (-0.984 - 3.02i)T + (-97.8 + 71.1i)T^{2}
13 1+(−1.52−3.00i)T+(−99.3+136.i)T2 1 + (-1.52 - 3.00i)T + (-99.3 + 136. i)T^{2}
17 1+(2.34+14.7i)T+(−274.+89.3i)T2 1 + (2.34 + 14.7i)T + (-274. + 89.3i)T^{2}
19 1+(−13.4+18.5i)T+(−111.−343.i)T2 1 + (-13.4 + 18.5i)T + (-111. - 343. i)T^{2}
23 1+(13.8+7.04i)T+(310.+427.i)T2 1 + (13.8 + 7.04i)T + (310. + 427. i)T^{2}
29 1+(−10.2−14.1i)T+(−259.+799.i)T2 1 + (-10.2 - 14.1i)T + (-259. + 799. i)T^{2}
31 1+(−9.99−7.25i)T+(296.+913.i)T2 1 + (-9.99 - 7.25i)T + (296. + 913. i)T^{2}
37 1+(−0.734+0.373i)T+(804.−1.10e3i)T2 1 + (-0.734 + 0.373i)T + (804. - 1.10e3i)T^{2}
41 1+(8.67−26.7i)T+(−1.35e3−988.i)T2 1 + (8.67 - 26.7i)T + (-1.35e3 - 988. i)T^{2}
43 1+(42.9−42.9i)T−1.84e3iT2 1 + (42.9 - 42.9i)T - 1.84e3iT^{2}
47 1+(−46.4−7.34i)T+(2.10e3+682.i)T2 1 + (-46.4 - 7.34i)T + (2.10e3 + 682. i)T^{2}
53 1+(0.616−3.89i)T+(−2.67e3−868.i)T2 1 + (0.616 - 3.89i)T + (-2.67e3 - 868. i)T^{2}
59 1+(−77.9−25.3i)T+(2.81e3+2.04e3i)T2 1 + (-77.9 - 25.3i)T + (2.81e3 + 2.04e3i)T^{2}
61 1+(−15.8−48.7i)T+(−3.01e3+2.18e3i)T2 1 + (-15.8 - 48.7i)T + (-3.01e3 + 2.18e3i)T^{2}
67 1+(−11.9−75.3i)T+(−4.26e3+1.38e3i)T2 1 + (-11.9 - 75.3i)T + (-4.26e3 + 1.38e3i)T^{2}
71 1+(−76.2+55.3i)T+(1.55e3−4.79e3i)T2 1 + (-76.2 + 55.3i)T + (1.55e3 - 4.79e3i)T^{2}
73 1+(101.+51.5i)T+(3.13e3+4.31e3i)T2 1 + (101. + 51.5i)T + (3.13e3 + 4.31e3i)T^{2}
79 1+(47.6+65.6i)T+(−1.92e3+5.93e3i)T2 1 + (47.6 + 65.6i)T + (-1.92e3 + 5.93e3i)T^{2}
83 1+(−68.6+10.8i)T+(6.55e3−2.12e3i)T2 1 + (-68.6 + 10.8i)T + (6.55e3 - 2.12e3i)T^{2}
89 1+(33.4−10.8i)T+(6.40e3−4.65e3i)T2 1 + (33.4 - 10.8i)T + (6.40e3 - 4.65e3i)T^{2}
97 1+(−4.45−0.705i)T+(8.94e3+2.90e3i)T2 1 + (-4.45 - 0.705i)T + (8.94e3 + 2.90e3i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−17.77269721662134697504269848025, −16.13426490551208514013979157018, −14.74582716221486166377770890154, −13.43091884764077174760655272028, −12.07069510386075968658512467034, −11.49800322540929296285759227246, −8.519293910065167974980853494833, −7.42103830341100601788438132572, −6.66081862542100131176156565699, −2.74885216966418794152214705996, 3.83355975003627897744198810923, 5.46419068157988654446933675120, 8.329302941153338822005442376376, 9.723599696420477027562014881655, 10.72647457273992502234395117389, 11.83620797045898702595751049969, 14.18170209941814146718697479075, 15.32785479620744146128573703619, 15.92854892807496740964262716767, 17.00220438349511852264715183018

Graph of the ZZ-function along the critical line