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Properties

Label 2-350-35.4-c3-0-26
Degree $2$
Conductor $350$
Sign $-0.999 + 0.0105i$
Analytic cond. $20.6506$
Root an. cond. $4.54430$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − i)2-s + (−8.22 − 4.74i)3-s + (1.99 − 3.46i)4-s − 18.9·6-s + (14.6 − 11.3i)7-s − 7.99i·8-s + (31.6 + 54.7i)9-s + (35.9 − 62.2i)11-s + (−32.9 + 18.9i)12-s − 70.4i·13-s + (14.0 − 34.2i)14-s + (−8 − 13.8i)16-s + (−21.9 − 12.6i)17-s + (109. + 63.2i)18-s + (35.4 + 61.3i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−1.58 − 0.914i)3-s + (0.249 − 0.433i)4-s − 1.29·6-s + (0.791 − 0.611i)7-s − 0.353i·8-s + (1.17 + 2.02i)9-s + (0.984 − 1.70i)11-s + (−0.791 + 0.457i)12-s − 1.50i·13-s + (0.268 − 0.654i)14-s + (−0.125 − 0.216i)16-s + (−0.312 − 0.180i)17-s + (1.43 + 0.827i)18-s + (0.427 + 0.741i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0105i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.999 + 0.0105i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ -0.999 + 0.0105i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.588264720\)
\(L(\frac12)\) \(\approx\) \(1.588264720\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 + i)T \)
5 \( 1 \)
7 \( 1 + (-14.6 + 11.3i)T \)
good3 \( 1 + (8.22 + 4.74i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (-35.9 + 62.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 70.4iT - 2.19e3T^{2} \)
17 \( 1 + (21.9 + 12.6i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-35.4 - 61.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-104. + 60.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 13.7T + 2.43e4T^{2} \)
31 \( 1 + (122. - 211. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (64.4 - 37.2i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 90.8T + 6.89e4T^{2} \)
43 \( 1 + 9.31iT - 7.95e4T^{2} \)
47 \( 1 + (241. - 139. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-79.6 - 45.9i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-18.9 + 32.8i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (96.4 + 167. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-372. - 214. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 76.0T + 3.57e5T^{2} \)
73 \( 1 + (691. + 399. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (5.52 + 9.56i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 603. iT - 5.71e5T^{2} \)
89 \( 1 + (-400. - 693. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 44.6iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.95839703977356289254381540024, −10.35399197797586672760561048062, −8.510146750635724888753451043665, −7.43461659774156952027844840896, −6.46332224620233137695826970608, −5.63461751720446135463286363667, −4.90406579734082603868007264753, −3.39119550009628980306570933028, −1.38180387579222802012250044737, −0.62645152078760175286861313965, 1.75443493742997297035414299217, 4.07216687548599378693197317639, 4.65557865122520399148933885306, 5.42461514575374400985517526859, 6.58333675327509960815453002062, 7.18794543996063109737315192237, 9.091466325425437074596881301313, 9.631348546287717434183420862360, 10.98743036377817048951878604756, 11.68329633273827898711513566433

Graph of the $Z$-function along the critical line