Properties

Label 2-350-25.21-c1-0-4
Degree $2$
Conductor $350$
Sign $-0.485 - 0.873i$
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  + (−0.309 + 0.951i)2-s + (2.59 + 1.88i)3-s + (−0.809 − 0.587i)4-s + (−0.0648 + 2.23i)5-s + (−2.59 + 1.88i)6-s + 7-s + (0.809 − 0.587i)8-s + (2.25 + 6.93i)9-s + (−2.10 − 0.752i)10-s + (1.17 − 3.61i)11-s + (−0.991 − 3.05i)12-s + (−1.68 − 5.19i)13-s + (−0.309 + 0.951i)14-s + (−4.38 + 5.67i)15-s + (0.309 + 0.951i)16-s + (−4.75 + 3.45i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (1.49 + 1.08i)3-s + (−0.404 − 0.293i)4-s + (−0.0290 + 0.999i)5-s + (−1.05 + 0.769i)6-s + 0.377·7-s + (0.286 − 0.207i)8-s + (0.751 + 2.31i)9-s + (−0.665 − 0.237i)10-s + (0.353 − 1.08i)11-s + (−0.286 − 0.880i)12-s + (−0.467 − 1.44i)13-s + (−0.0825 + 0.254i)14-s + (−1.13 + 1.46i)15-s + (0.0772 + 0.237i)16-s + (−1.15 + 0.838i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.485 - 0.873i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.485 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945068 + 1.60680i\)
\(L(\frac12)\) \(\approx\) \(0.945068 + 1.60680i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.0648 - 2.23i)T \)
7 \( 1 - T \)
good3 \( 1 + (-2.59 - 1.88i)T + (0.927 + 2.85i)T^{2} \)
11 \( 1 + (-1.17 + 3.61i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.68 + 5.19i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.75 - 3.45i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.35 + 3.16i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.697 + 2.14i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.79 + 2.75i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.13 + 3.00i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.131 - 0.403i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.25 - 6.94i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.16T + 43T^{2} \)
47 \( 1 + (0.214 + 0.155i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-5.20 - 3.78i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.61 - 8.05i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.57 + 4.84i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (9.86 - 7.16i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (6.35 + 4.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.54 + 7.83i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.00 - 3.63i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.35 + 4.61i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.135 + 0.415i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.44 + 2.49i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.34383941863628666312167099818, −10.54449912123751417524790776228, −9.852675579007510608821084989579, −8.860909661094900073241122514330, −8.149617429842308529848986288802, −7.41083729468156755142943770550, −6.01094356017390503433603536836, −4.69774449956730408865257029898, −3.52967230830175752564534802181, −2.61587443992294497052645453710, 1.48977121541007538470471186829, 2.21157317336748541989769452712, 3.79085636468112921047059844316, 4.83597711736525823848262779733, 6.89887388710654294646050136710, 7.53402097218056876521723111310, 8.603578701666338627670254413141, 9.216256231014783503242499841404, 9.756510956519315093906089706419, 11.70763856785256982190941591961

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