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Properties

Label 2-350-35.4-c3-0-32
Degree $2$
Conductor $350$
Sign $-0.433 + 0.901i$
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 + (2.69 + 1.55i)3-s + (1.99 − 3.46i)4-s + 6.22·6-s + (−3.64 − 18.1i)7-s − 7.99i·8-s + (−8.65 − 14.9i)9-s + (2.50 − 4.33i)11-s + (10.7 − 6.22i)12-s − 2.87i·13-s + (−24.4 − 27.8i)14-s + (−8 − 13.8i)16-s + (−40.9 − 23.6i)17-s + (−29.9 − 17.3i)18-s + (−11.8 − 20.5i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.518 + 0.299i)3-s + (0.249 − 0.433i)4-s + 0.423·6-s + (−0.196 − 0.980i)7-s − 0.353i·8-s + (−0.320 − 0.555i)9-s + (0.0686 − 0.118i)11-s + (0.259 − 0.149i)12-s − 0.0612i·13-s + (−0.467 − 0.530i)14-s + (−0.125 − 0.216i)16-s + (−0.584 − 0.337i)17-s + (−0.392 − 0.226i)18-s + (−0.143 − 0.247i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.433 + 0.901i$
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: \((2,\ 350,\ (\ :3/2),\ -0.433 + 0.901i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.471508046\)
\(L(\frac12)\) \(\approx\) \(2.471508046\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.64 + 18.1i)T \)
good3 \( 1 + (-2.69 - 1.55i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (-2.50 + 4.33i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 2.87iT - 2.19e3T^{2} \)
17 \( 1 + (40.9 + 23.6i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (11.8 + 20.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (56.9 - 32.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 91.7T + 2.43e4T^{2} \)
31 \( 1 + (-64.7 + 112. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-321. + 185. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 19.0T + 6.89e4T^{2} \)
43 \( 1 - 117. iT - 7.95e4T^{2} \)
47 \( 1 + (-519. + 300. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-151. - 87.4i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (24.4 - 42.3i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (349. + 605. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-507. - 292. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 756.T + 3.57e5T^{2} \)
73 \( 1 + (-709. - 409. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-240. - 417. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 269. iT - 5.71e5T^{2} \)
89 \( 1 + (-73.2 - 126. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 241. iT - 9.12e5T^{2} \)
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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

−10.87347258691135091814145322099, −9.846929784044648649279465937525, −9.150880625805758377612994702439, −7.902757469959563118101105029844, −6.81681005079271801670558807532, −5.80488989609194932310908803881, −4.36051780448400320921741677054, −3.66230014180179189698990545522, −2.45206437714306830135966601719, −0.63293263300982021695522931640, 2.02907611574330201579817240880, 2.97236354238032005557101287477, 4.39529276369085131463827892114, 5.56692840264421250732188273670, 6.41950869203143063756103987405, 7.61290909120712115743578092427, 8.441406636708816749539436744961, 9.215394058238227525911087080831, 10.55079763793386487664023088243, 11.58387578648222530772321394038

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