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Properties

Label 2-350-175.108-c1-0-15
Degree 22
Conductor 350350
Sign −0.895+0.445i-0.895 + 0.445i
Analytic cond. 2.794762.79476
Root an. cond. 1.671751.67175
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  + (−0.0523 − 0.998i)2-s + (−0.158 − 0.128i)3-s + (−0.994 + 0.104i)4-s + (−1.20 + 1.88i)5-s + (−0.119 + 0.164i)6-s + (0.823 − 2.51i)7-s + (0.156 + 0.987i)8-s + (−0.615 − 2.89i)9-s + (1.94 + 1.10i)10-s + (−4.28 − 0.911i)11-s + (0.170 + 0.110i)12-s + (−1.59 − 3.13i)13-s + (−2.55 − 0.690i)14-s + (0.432 − 0.143i)15-s + (0.978 − 0.207i)16-s + (3.12 + 1.20i)17-s + ⋯
L(s)  = 1  + (−0.0370 − 0.706i)2-s + (−0.0913 − 0.0739i)3-s + (−0.497 + 0.0522i)4-s + (−0.539 + 0.842i)5-s + (−0.0488 + 0.0672i)6-s + (0.311 − 0.950i)7-s + (0.0553 + 0.349i)8-s + (−0.205 − 0.964i)9-s + (0.614 + 0.349i)10-s + (−1.29 − 0.274i)11-s + (0.0492 + 0.0320i)12-s + (−0.443 − 0.870i)13-s + (−0.682 − 0.184i)14-s + (0.111 − 0.0370i)15-s + (0.244 − 0.0519i)16-s + (0.758 + 0.291i)17-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=((−0.895+0.445i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+1/2) L(s)=((−0.895+0.445i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: −0.895+0.445i-0.895 + 0.445i
Analytic conductor: 2.794762.79476
Root analytic conductor: 1.671751.67175
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ350(283,⋅)\chi_{350} (283, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 350, ( :1/2), −0.895+0.445i)(2,\ 350,\ (\ :1/2),\ -0.895 + 0.445i)

Particular Values

L(1)L(1) ≈\approx 0.164208−0.697875i0.164208 - 0.697875i
L(12)L(\frac12) ≈\approx 0.164208−0.697875i0.164208 - 0.697875i
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)
bad2 1+(0.0523+0.998i)T 1 + (0.0523 + 0.998i)T
5 1+(1.20−1.88i)T 1 + (1.20 - 1.88i)T
7 1+(−0.823+2.51i)T 1 + (-0.823 + 2.51i)T
good3 1+(0.158+0.128i)T+(0.623+2.93i)T2 1 + (0.158 + 0.128i)T + (0.623 + 2.93i)T^{2}
11 1+(4.28+0.911i)T+(10.0+4.47i)T2 1 + (4.28 + 0.911i)T + (10.0 + 4.47i)T^{2}
13 1+(1.59+3.13i)T+(−7.64+10.5i)T2 1 + (1.59 + 3.13i)T + (-7.64 + 10.5i)T^{2}
17 1+(−3.12−1.20i)T+(12.6+11.3i)T2 1 + (-3.12 - 1.20i)T + (12.6 + 11.3i)T^{2}
19 1+(−0.644+6.13i)T+(−18.5−3.95i)T2 1 + (-0.644 + 6.13i)T + (-18.5 - 3.95i)T^{2}
23 1+(5.32−0.279i)T+(22.8−2.40i)T2 1 + (5.32 - 0.279i)T + (22.8 - 2.40i)T^{2}
29 1+(−0.478−0.658i)T+(−8.96+27.5i)T2 1 + (-0.478 - 0.658i)T + (-8.96 + 27.5i)T^{2}
31 1+(0.875+1.96i)T+(−20.7+23.0i)T2 1 + (0.875 + 1.96i)T + (-20.7 + 23.0i)T^{2}
37 1+(4.88−7.52i)T+(−15.0−33.8i)T2 1 + (4.88 - 7.52i)T + (-15.0 - 33.8i)T^{2}
41 1+(−10.9−3.54i)T+(33.1+24.0i)T2 1 + (-10.9 - 3.54i)T + (33.1 + 24.0i)T^{2}
43 1+(−5.14−5.14i)T+43iT2 1 + (-5.14 - 5.14i)T + 43iT^{2}
47 1+(0.179+0.466i)T+(−34.9+31.4i)T2 1 + (0.179 + 0.466i)T + (-34.9 + 31.4i)T^{2}
53 1+(−1.91+2.36i)T+(−11.0−51.8i)T2 1 + (-1.91 + 2.36i)T + (-11.0 - 51.8i)T^{2}
59 1+(5.51+6.12i)T+(−6.16+58.6i)T2 1 + (5.51 + 6.12i)T + (-6.16 + 58.6i)T^{2}
61 1+(1.23+1.11i)T+(6.37+60.6i)T2 1 + (1.23 + 1.11i)T + (6.37 + 60.6i)T^{2}
67 1+(1.25−3.27i)T+(−49.7−44.8i)T2 1 + (1.25 - 3.27i)T + (-49.7 - 44.8i)T^{2}
71 1+(−12.1+8.79i)T+(21.9−67.5i)T2 1 + (-12.1 + 8.79i)T + (21.9 - 67.5i)T^{2}
73 1+(−2.61+1.69i)T+(29.6−66.6i)T2 1 + (-2.61 + 1.69i)T + (29.6 - 66.6i)T^{2}
79 1+(−6.08+13.6i)T+(−52.8−58.7i)T2 1 + (-6.08 + 13.6i)T + (-52.8 - 58.7i)T^{2}
83 1+(−7.78+1.23i)T+(78.9−25.6i)T2 1 + (-7.78 + 1.23i)T + (78.9 - 25.6i)T^{2}
89 1+(2.00−2.22i)T+(−9.30−88.5i)T2 1 + (2.00 - 2.22i)T + (-9.30 - 88.5i)T^{2}
97 1+(−9.90−1.56i)T+(92.2+29.9i)T2 1 + (-9.90 - 1.56i)T + (92.2 + 29.9i)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

−10.97447352497791110208646211509, −10.43788675054006754624639800391, −9.567636154999673652608537545374, −8.055605125371535414888293016604, −7.56274708333197450875140207837, −6.26658151321016606812493660825, −4.91344417845436477742848042178, −3.62176201304653677979425423526, −2.75501033995479605733207944293, −0.49610368408848332259000542041, 2.17540173613874137228653319327, 4.14139238531478748435080463920, 5.26544021929230551221162455033, 5.68150045132225509629149031978, 7.55098469161692268510041689085, 7.942971339930223665732091864090, 8.878099572142757960605538556824, 9.866094930475126449969670415051, 10.94562101051338524722795942523, 12.27392421788919938209993889966

Graph of the ZZ-function along the critical line