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Properties

Label 2-350-35.4-c3-0-4
Degree $2$
Conductor $350$
Sign $0.00780 - 0.999i$
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 + (−2.66 − 1.53i)3-s + (1.99 − 3.46i)4-s − 6.15·6-s + (11.0 + 14.8i)7-s − 7.99i·8-s + (−8.76 − 15.1i)9-s + (−30.6 + 53.0i)11-s + (−10.6 + 6.15i)12-s + 23.4i·13-s + (33.9 + 14.6i)14-s + (−8 − 13.8i)16-s + (−117. − 67.8i)17-s + (−30.3 − 17.5i)18-s + (31.0 + 53.7i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.512 − 0.296i)3-s + (0.249 − 0.433i)4-s − 0.418·6-s + (0.596 + 0.802i)7-s − 0.353i·8-s + (−0.324 − 0.562i)9-s + (−0.839 + 1.45i)11-s + (−0.256 + 0.148i)12-s + 0.501i·13-s + (0.649 + 0.280i)14-s + (−0.125 − 0.216i)16-s + (−1.67 − 0.968i)17-s + (−0.397 − 0.229i)18-s + (0.374 + 0.649i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00780 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00780 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.00780 - 0.999i$
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: \((2,\ 350,\ (\ :3/2),\ 0.00780 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.067988261\)
\(L(\frac12)\) \(\approx\) \(1.067988261\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.0 - 14.8i)T \)
good3 \( 1 + (2.66 + 1.53i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (30.6 - 53.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 23.4iT - 2.19e3T^{2} \)
17 \( 1 + (117. + 67.8i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-31.0 - 53.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (63.9 - 36.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 303.T + 2.43e4T^{2} \)
31 \( 1 + (86.1 - 149. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (330. - 190. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 163.T + 6.89e4T^{2} \)
43 \( 1 - 174. iT - 7.95e4T^{2} \)
47 \( 1 + (195. - 112. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-168. - 97.3i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-36.0 + 62.3i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-81.7 - 141. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-99.1 - 57.2i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 442.T + 3.57e5T^{2} \)
73 \( 1 + (697. + 402. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-129. - 224. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.06e3iT - 5.71e5T^{2} \)
89 \( 1 + (518. + 898. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 749. iT - 9.12e5T^{2} \)
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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

−11.65744030535595826326559727038, −10.59271070810770660953800734869, −9.561567129474017502001032698522, −8.567736355164676054508429822596, −7.20708571547453256141686810025, −6.37737559696661014830052116735, −5.19305170150202526001148729137, −4.54057213047041354312607069345, −2.79683731549530249380744212340, −1.69347009191219250282127964404, 0.30616489761604385823315638464, 2.46868731518455627730339693384, 3.92971350904434226546844642926, 4.94438103190118913934544892713, 5.75393267179840864343040790161, 6.80368721571556215893975541101, 8.080454489385349347855403746140, 8.552449248232460125443234386855, 10.42919905799239251542503726837, 10.86312231234827084162391080451

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