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Properties

Label 2-325-13.9-c1-0-8
Degree 22
Conductor 325325
Sign 0.548−0.835i0.548 - 0.835i
Analytic cond. 2.595132.59513
Root an. cond. 1.610941.61094
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.134 + 0.232i)2-s + (0.301 − 0.522i)3-s + (0.963 + 1.66i)4-s + (0.0810 + 0.140i)6-s + (0.715 + 1.23i)7-s − 1.05·8-s + (1.31 + 2.28i)9-s + (0.0810 − 0.140i)11-s + 1.16·12-s + (−2.41 − 2.67i)13-s − 0.384·14-s + (−1.78 + 3.09i)16-s + (1.41 + 2.44i)17-s − 0.708·18-s + (1.96 + 3.40i)19-s + ⋯
L(s)  = 1  + (−0.0950 + 0.164i)2-s + (0.174 − 0.301i)3-s + (0.481 + 0.834i)4-s + (0.0330 + 0.0573i)6-s + (0.270 + 0.468i)7-s − 0.373·8-s + (0.439 + 0.761i)9-s + (0.0244 − 0.0423i)11-s + 0.335·12-s + (−0.670 − 0.742i)13-s − 0.102·14-s + (−0.446 + 0.773i)16-s + (0.342 + 0.592i)17-s − 0.167·18-s + (0.450 + 0.780i)19-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=((0.548−0.835i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+1/2) L(s)=((0.548−0.835i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: 0.548−0.835i0.548 - 0.835i
Analytic conductor: 2.595132.59513
Root analytic conductor: 1.610941.61094
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ325(126,⋅)\chi_{325} (126, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 325, ( :1/2), 0.548−0.835i)(2,\ 325,\ (\ :1/2),\ 0.548 - 0.835i)

Particular Values

L(1)L(1) ≈\approx 1.30850+0.706297i1.30850 + 0.706297i
L(12)L(\frac12) ≈\approx 1.30850+0.706297i1.30850 + 0.706297i
L(32)L(\frac{3}{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 1
13 1+(2.41+2.67i)T 1 + (2.41 + 2.67i)T
good2 1+(0.134−0.232i)T+(−1−1.73i)T2 1 + (0.134 - 0.232i)T + (-1 - 1.73i)T^{2}
3 1+(−0.301+0.522i)T+(−1.5−2.59i)T2 1 + (-0.301 + 0.522i)T + (-1.5 - 2.59i)T^{2}
7 1+(−0.715−1.23i)T+(−3.5+6.06i)T2 1 + (-0.715 - 1.23i)T + (-3.5 + 6.06i)T^{2}
11 1+(−0.0810+0.140i)T+(−5.5−9.52i)T2 1 + (-0.0810 + 0.140i)T + (-5.5 - 9.52i)T^{2}
17 1+(−1.41−2.44i)T+(−8.5+14.7i)T2 1 + (-1.41 - 2.44i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.96−3.40i)T+(−9.5+16.4i)T2 1 + (-1.96 - 3.40i)T + (-9.5 + 16.4i)T^{2}
23 1+(−2.36+4.09i)T+(−11.5−19.9i)T2 1 + (-2.36 + 4.09i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.99+3.45i)T+(−14.5−25.1i)T2 1 + (-1.99 + 3.45i)T + (-14.5 - 25.1i)T^{2}
31 1+0.453T+31T2 1 + 0.453T + 31T^{2}
37 1+(2.52−4.36i)T+(−18.5−32.0i)T2 1 + (2.52 - 4.36i)T + (-18.5 - 32.0i)T^{2}
41 1+(−4.29+7.43i)T+(−20.5−35.5i)T2 1 + (-4.29 + 7.43i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.33−4.03i)T+(−21.5+37.2i)T2 1 + (-2.33 - 4.03i)T + (-21.5 + 37.2i)T^{2}
47 1+11.4T+47T2 1 + 11.4T + 47T^{2}
53 1−7.30T+53T2 1 - 7.30T + 53T^{2}
59 1+(4.98+8.63i)T+(−29.5+51.0i)T2 1 + (4.98 + 8.63i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.726+1.25i)T+(−30.5+52.8i)T2 1 + (0.726 + 1.25i)T + (-30.5 + 52.8i)T^{2}
67 1+(−3.17+5.50i)T+(−33.5−58.0i)T2 1 + (-3.17 + 5.50i)T + (-33.5 - 58.0i)T^{2}
71 1+(7.02+12.1i)T+(−35.5+61.4i)T2 1 + (7.02 + 12.1i)T + (-35.5 + 61.4i)T^{2}
73 1+7.75T+73T2 1 + 7.75T + 73T^{2}
79 1+11.2T+79T2 1 + 11.2T + 79T^{2}
83 1−9.45T+83T2 1 - 9.45T + 83T^{2}
89 1+(−4.33+7.50i)T+(−44.5−77.0i)T2 1 + (-4.33 + 7.50i)T + (-44.5 - 77.0i)T^{2}
97 1+(7.40+12.8i)T+(−48.5+84.0i)T2 1 + (7.40 + 12.8i)T + (-48.5 + 84.0i)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

−11.97326256028608982065232159854, −10.85277327539404454988032916168, −9.970862927052501390850645376729, −8.570257900094381620791207790329, −7.947163007507300041065290417534, −7.19787450661979239667536816647, −6.00036447364893754614782741171, −4.72953985306795719842202753117, −3.21827616564514736408726341938, −2.04691440609770714909605607105, 1.22249992441489060596500792987, 2.88896443726096609350629779134, 4.38253418394432217872515640317, 5.43033595061470435161501801932, 6.79219310570615819398789212897, 7.35418158910286248817041685081, 9.066455423612529080995609951809, 9.605578193179933896889976380921, 10.46607022218642748003500768551, 11.43021679818646908695468425357

Graph of the ZZ-function along the critical line