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Properties

Label 2-465-15.2-c1-0-31
Degree $2$
Conductor $465$
Sign $0.917 + 0.398i$
Analytic cond. $3.71304$
Root an. cond. $1.92692$
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  + (−0.195 + 0.195i)2-s + (−1.67 + 0.423i)3-s + 1.92i·4-s + (2.09 − 0.787i)5-s + (0.245 − 0.411i)6-s + (−2.75 − 2.75i)7-s + (−0.767 − 0.767i)8-s + (2.64 − 1.42i)9-s + (−0.255 + 0.563i)10-s − 1.30i·11-s + (−0.815 − 3.23i)12-s + (3.21 − 3.21i)13-s + 1.07·14-s + (−3.18 + 2.20i)15-s − 3.54·16-s + (1.44 − 1.44i)17-s + ⋯
L(s)  = 1  + (−0.138 + 0.138i)2-s + (−0.969 + 0.244i)3-s + 0.961i·4-s + (0.935 − 0.352i)5-s + (0.100 − 0.167i)6-s + (−1.04 − 1.04i)7-s + (−0.271 − 0.271i)8-s + (0.880 − 0.474i)9-s + (−0.0806 + 0.178i)10-s − 0.392i·11-s + (−0.235 − 0.932i)12-s + (0.892 − 0.892i)13-s + 0.288·14-s + (−0.821 + 0.570i)15-s − 0.886·16-s + (0.349 − 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.917 + 0.398i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.917 + 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.954249 - 0.198436i\)
\(L(\frac12)\) \(\approx\) \(0.954249 - 0.198436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 - 0.423i)T \)
5 \( 1 + (-2.09 + 0.787i)T \)
31 \( 1 + T \)
good2 \( 1 + (0.195 - 0.195i)T - 2iT^{2} \)
7 \( 1 + (2.75 + 2.75i)T + 7iT^{2} \)
11 \( 1 + 1.30iT - 11T^{2} \)
13 \( 1 + (-3.21 + 3.21i)T - 13iT^{2} \)
17 \( 1 + (-1.44 + 1.44i)T - 17iT^{2} \)
19 \( 1 - 1.91iT - 19T^{2} \)
23 \( 1 + (-0.841 - 0.841i)T + 23iT^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
37 \( 1 + (2.30 + 2.30i)T + 37iT^{2} \)
41 \( 1 + 2.71iT - 41T^{2} \)
43 \( 1 + (-1.08 + 1.08i)T - 43iT^{2} \)
47 \( 1 + (-6.09 + 6.09i)T - 47iT^{2} \)
53 \( 1 + (2.73 + 2.73i)T + 53iT^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 + (-5.05 - 5.05i)T + 67iT^{2} \)
71 \( 1 + 13.0iT - 71T^{2} \)
73 \( 1 + (5.56 - 5.56i)T - 73iT^{2} \)
79 \( 1 - 7.10iT - 79T^{2} \)
83 \( 1 + (-2.09 - 2.09i)T + 83iT^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + (0.122 + 0.122i)T + 97iT^{2} \)
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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

−10.73325148937947564107941500115, −10.22321762611588187575980386155, −9.324825998020989341449527283406, −8.289069787746136426632570521948, −7.06900546135267640602251886936, −6.37394708203707649826245402016, −5.44201765171353710654497675058, −4.10235251067272812666174289909, −3.14430057988357194615309757616, −0.793642691968699249747518798817, 1.41544881013748721086700329618, 2.66301608649013182199186780336, 4.67929762941735518458749729224, 5.76484767893206802372164305687, 6.28164774113073492371918818116, 6.85582909129389921439421030727, 8.770721599641371774568473250182, 9.545400263378908374883477393284, 10.21938717357852880599898784994, 10.96013479669725845697237918813

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