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Properties

Label 2-350-35.9-c3-0-18
Degree $2$
Conductor $350$
Sign $0.804 + 0.593i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.804 + 0.593i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ 0.804 + 0.593i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.449837995\)
\(L(\frac12)\) \(\approx\) \(1.449837995\)
\(L(\frac{5}{2})\) 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

−11.04443115208330383757291670824, −9.863317924129017751303913143868, −9.096914933418415449059404842579, −8.220275653298174654284442630632, −7.54834351707089825643397190641, −6.16148172366104122196428011340, −5.14055308362178996588753831729, −3.05836079014654297638732170770, −2.70954283907476038645091239171, −0.816690550918270924814262721592, 0.935635760293567446139789441360, 2.76942634926891023449585296023, 3.93447232618439945811857439431, 5.35952103702477668688895784792, 6.53606810965746541202926487688, 7.57276646693962628419998685059, 8.217516451314000965590071006281, 9.473676925345233194380193666814, 10.09525307431992780665558208580, 10.61450981587110597935863366245

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