Properties

Label 2-350-25.4-c1-0-15
Degree $2$
Conductor $350$
Sign $0.554 + 0.832i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 + 0.309i)2-s + (1.41 − 1.95i)3-s + (0.809 + 0.587i)4-s + (−0.0248 − 2.23i)5-s + (1.95 − 1.41i)6-s i·7-s + (0.587 + 0.809i)8-s + (−0.869 − 2.67i)9-s + (0.667 − 2.13i)10-s + (−1.92 + 5.92i)11-s + (2.29 − 0.745i)12-s + (−0.224 + 0.0730i)13-s + (0.309 − 0.951i)14-s + (−4.39 − 3.12i)15-s + (0.309 + 0.951i)16-s + (−2.50 − 3.45i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.818 − 1.12i)3-s + (0.404 + 0.293i)4-s + (−0.0110 − 0.999i)5-s + (0.796 − 0.578i)6-s − 0.377i·7-s + (0.207 + 0.286i)8-s + (−0.289 − 0.891i)9-s + (0.211 − 0.674i)10-s + (−0.580 + 1.78i)11-s + (0.661 − 0.215i)12-s + (−0.0623 + 0.0202i)13-s + (0.0825 − 0.254i)14-s + (−1.13 − 0.805i)15-s + (0.0772 + 0.237i)16-s + (−0.608 − 0.836i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.554 + 0.832i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11499 - 1.13234i\)
\(L(\frac12)\) \(\approx\) \(2.11499 - 1.13234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (0.0248 + 2.23i)T \)
7 \( 1 + iT \)
good3 \( 1 + (-1.41 + 1.95i)T + (-0.927 - 2.85i)T^{2} \)
11 \( 1 + (1.92 - 5.92i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.224 - 0.0730i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.50 + 3.45i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.0269 - 0.0196i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-6.11 - 1.98i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-6.08 - 4.42i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.734 - 0.533i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (5.22 - 1.69i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.968 - 2.98i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.86iT - 43T^{2} \)
47 \( 1 + (-2.84 + 3.91i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.0640 - 0.0881i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.10 + 3.40i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.02 - 12.4i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-4.52 - 6.23i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (2.17 + 1.57i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.97 + 2.91i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.2 + 7.43i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.73 + 6.51i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.548 - 1.68i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (10.4 - 14.4i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.91581534705421337627238409206, −10.42018159889386398602754775656, −9.211134461060755076807851534950, −8.357561761162818176775046769358, −7.21920553720596496088168861864, −7.02904824780169914636631523083, −5.22181742808560457982068595128, −4.45304042161044704803652794726, −2.77903381813458086475735499788, −1.57947887734668845533947080587, 2.69008537940925355912501018263, 3.25896346614852320956102292116, 4.34829264760378619490999446457, 5.65410124550791123174889079536, 6.59404201667781997463020763047, 8.108684774827147596343091504488, 8.876924148695165005747025563132, 10.00622457268573801445765523660, 10.84293237535512072862816123718, 11.22478648303391219227634716303

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