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Properties

Label 2-350-7.2-c3-0-22
Degree $2$
Conductor $350$
Sign $0.905 - 0.424i$
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 + 1.73i)2-s + (3.00 + 5.19i)3-s + (−1.99 − 3.46i)4-s − 12.0·6-s + (11.5 − 14.4i)7-s + 7.99·8-s + (−4.51 + 7.81i)9-s + (−4.57 − 7.92i)11-s + (12.0 − 20.7i)12-s + 32.7·13-s + (13.5 + 34.4i)14-s + (−8 + 13.8i)16-s + (−54.8 − 95.0i)17-s + (−9.02 − 15.6i)18-s + (11.1 − 19.2i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.577 + 1.00i)3-s + (−0.249 − 0.433i)4-s − 0.816·6-s + (0.622 − 0.782i)7-s + 0.353·8-s + (−0.167 + 0.289i)9-s + (−0.125 − 0.217i)11-s + (0.288 − 0.500i)12-s + 0.697·13-s + (0.259 + 0.657i)14-s + (−0.125 + 0.216i)16-s + (−0.783 − 1.35i)17-s + (−0.118 − 0.204i)18-s + (0.134 − 0.232i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.905 - 0.424i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ 0.905 - 0.424i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.980387551\)
\(L(\frac12)\) \(\approx\) \(1.980387551\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
5 \( 1 \)
7 \( 1 + (-11.5 + 14.4i)T \)
good3 \( 1 + (-3.00 - 5.19i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (4.57 + 7.92i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 32.7T + 2.19e3T^{2} \)
17 \( 1 + (54.8 + 95.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-11.1 + 19.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-95.6 + 165. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 190.T + 2.43e4T^{2} \)
31 \( 1 + (17.8 + 30.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (28.8 - 50.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 39.7T + 6.89e4T^{2} \)
43 \( 1 + 323.T + 7.95e4T^{2} \)
47 \( 1 + (28.5 - 49.4i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-264. - 458. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-383. - 663. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (262. - 453. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (185. + 321. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 722.T + 3.57e5T^{2} \)
73 \( 1 + (414. + 717. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (247. - 428. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.24e3T + 5.71e5T^{2} \)
89 \( 1 + (10.3 - 17.9i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.01e3T + 9.12e5T^{2} \)
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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

−10.70936303881757620740739596418, −10.21467853584548882673107618594, −8.997234341733529413018742824511, −8.612886899352652967531222717872, −7.39471879329448235735296226346, −6.47715457443770717500112628764, −4.90312977940982228424682674453, −4.31771082732174294035809433257, −2.89634721364064842607341954515, −0.827957514677913104210808830651, 1.37823991341442080248500559309, 2.13731181582994261534702621659, 3.44356088565747086883937134886, 4.96545897790860911554059657245, 6.33822280383239817503200513854, 7.47171343346742732784898110501, 8.403658341754394665946598789503, 8.794678888222173261658927112782, 10.11576620774572003961815479225, 11.13255022540024185227308777658

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