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Properties

Label 2-350-35.4-c3-0-35
Degree $2$
Conductor $350$
Sign $-0.652 - 0.758i$
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 + (−4.33 − 2.5i)3-s + (1.99 − 3.46i)4-s − 10·6-s + (−12.1 − 14i)7-s − 7.99i·8-s + (−0.999 − 1.73i)9-s + (28.5 − 49.3i)11-s + (−17.3 + 10i)12-s + 70i·13-s + (−35 − 12.1i)14-s + (−8 − 13.8i)16-s + (−44.1 − 25.5i)17-s + (−3.46 − 1.99i)18-s + (2.5 + 4.33i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.833 − 0.481i)3-s + (0.249 − 0.433i)4-s − 0.680·6-s + (−0.654 − 0.755i)7-s − 0.353i·8-s + (−0.0370 − 0.0641i)9-s + (0.781 − 1.35i)11-s + (−0.416 + 0.240i)12-s + 1.49i·13-s + (−0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.630 − 0.363i)17-s + (−0.0453 − 0.0261i)18-s + (0.0301 + 0.0522i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.652 - 0.758i$
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: \((2,\ 350,\ (\ :3/2),\ -0.652 - 0.758i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5475444213\)
\(L(\frac12)\) \(\approx\) \(0.5475444213\)
\(L(\frac{5}{2})\) 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 \( 1 + (12.1 + 14i)T \)
good3 \( 1 + (4.33 + 2.5i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (-28.5 + 49.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 70iT - 2.19e3T^{2} \)
17 \( 1 + (44.1 + 25.5i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (59.7 - 34.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 114T + 2.43e4T^{2} \)
31 \( 1 + (11.5 - 19.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (219. - 126.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 42T + 6.89e4T^{2} \)
43 \( 1 - 124iT - 7.95e4T^{2} \)
47 \( 1 + (-174. + 100.5i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (340. + 196.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-109.5 + 189. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-354.5 - 614. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (362. + 209.5i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 96T + 3.57e5T^{2} \)
73 \( 1 + (271. + 156.5i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-230.5 - 399. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 588iT - 5.71e5T^{2} \)
89 \( 1 + (508.5 + 880. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.83e3iT - 9.12e5T^{2} \)
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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

−10.92045975016415343253913378631, −9.662272670767046749134335750044, −8.779871398660652383785952490827, −7.05676604149077927211198447220, −6.51174405210762568154017714429, −5.69869904720772361044094555039, −4.26150233728614343931018405293, −3.33501071928301406892396595896, −1.48145059734904319420535005946, −0.17167859070247344381296695529, 2.29701588817244983795108292152, 3.77542093803792467866303493012, 4.91072213859806664689677113174, 5.72550512494530664423676493850, 6.54002136731273171132352719477, 7.68884484308515035076529292454, 8.908085530846674386106316606541, 9.973063819686227560246458196653, 10.76290633393186984453220958935, 11.84968283284805330352607870257

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