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Properties

Label 2-350-25.4-c1-0-14
Degree $2$
Conductor $350$
Sign $0.445 + 0.895i$
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 + (1.91 − 2.63i)3-s + (0.809 + 0.587i)4-s + (−2.23 − 0.115i)5-s + (2.63 − 1.91i)6-s + i·7-s + (0.587 + 0.809i)8-s + (−2.34 − 7.20i)9-s + (−2.08 − 0.799i)10-s + (1.35 − 4.16i)11-s + (3.09 − 1.00i)12-s + (0.626 − 0.203i)13-s + (−0.309 + 0.951i)14-s + (−4.57 + 5.65i)15-s + (0.309 + 0.951i)16-s + (4.67 + 6.43i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (1.10 − 1.51i)3-s + (0.404 + 0.293i)4-s + (−0.998 − 0.0516i)5-s + (1.07 − 0.780i)6-s + 0.377i·7-s + (0.207 + 0.286i)8-s + (−0.780 − 2.40i)9-s + (−0.660 − 0.252i)10-s + (0.408 − 1.25i)11-s + (0.892 − 0.290i)12-s + (0.173 − 0.0564i)13-s + (−0.0825 + 0.254i)14-s + (−1.18 + 1.45i)15-s + (0.0772 + 0.237i)16-s + (1.13 + 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.445 + 0.895i$
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: \((2,\ 350,\ (\ :1/2),\ 0.445 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99905 - 1.23743i\)
\(L(\frac12)\) \(\approx\) \(1.99905 - 1.23743i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.115i)T \)
7 \( 1 - iT \)
good3 \( 1 + (-1.91 + 2.63i)T + (-0.927 - 2.85i)T^{2} \)
11 \( 1 + (-1.35 + 4.16i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.626 + 0.203i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.67 - 6.43i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.64 - 1.92i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.08 - 0.352i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.10 - 2.98i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (8.14 - 5.91i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (4.08 - 1.32i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.48 - 4.55i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.18iT - 43T^{2} \)
47 \( 1 + (-1.71 + 2.36i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.275 + 0.378i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.34 - 4.14i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.519 - 1.59i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (6.81 + 9.37i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-10.4 - 7.59i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.370 - 0.120i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.04 + 4.39i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.15 + 7.09i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.369 + 1.13i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-9.86 + 13.5i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.83153693624776651204193841310, −10.70646192110175605948576475459, −8.700606552030958694422886461599, −8.465397281197205831901994561884, −7.56227333937162235978614012285, −6.62581384702277928221497047769, −5.73534635525474677943174738017, −3.71643945579568321663618700754, −3.16271881785240487462699163528, −1.46992586244842166145323147562, 2.58142554093942619511615753495, 3.67205130395464832249945767937, 4.36090642275215016601733547750, 5.13782389935463424546624670729, 7.12226344313298470436127936799, 7.88717850028169364212211029587, 9.132007784853980867531614120125, 9.798685166907379069956901543057, 10.71591842169423090744008160026, 11.51353746041411060601735966328

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