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Properties

Label 2-945-35.13-c1-0-16
Degree $2$
Conductor $945$
Sign $-0.434 - 0.900i$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
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  + (0.515 − 0.515i)2-s + 1.46i·4-s + (−1.00 + 1.99i)5-s + (1.67 + 2.04i)7-s + (1.78 + 1.78i)8-s + (0.510 + 1.54i)10-s + 1.07·11-s + (−3.17 + 3.17i)13-s + (1.92 + 0.194i)14-s − 1.09·16-s + (−3.65 − 3.65i)17-s − 0.950·19-s + (−2.93 − 1.47i)20-s + (0.554 − 0.554i)22-s + (−0.550 − 0.550i)23-s + ⋯
L(s)  = 1  + (0.364 − 0.364i)2-s + 0.733i·4-s + (−0.450 + 0.892i)5-s + (0.632 + 0.774i)7-s + (0.632 + 0.632i)8-s + (0.161 + 0.489i)10-s + 0.323·11-s + (−0.880 + 0.880i)13-s + (0.513 + 0.0520i)14-s − 0.272·16-s + (−0.886 − 0.886i)17-s − 0.218·19-s + (−0.655 − 0.330i)20-s + (0.118 − 0.118i)22-s + (−0.114 − 0.114i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.434 - 0.900i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.434 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.820491 + 1.30612i\)
\(L(\frac12)\) \(\approx\) \(0.820491 + 1.30612i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.00 - 1.99i)T \)
7 \( 1 + (-1.67 - 2.04i)T \)
good2 \( 1 + (-0.515 + 0.515i)T - 2iT^{2} \)
11 \( 1 - 1.07T + 11T^{2} \)
13 \( 1 + (3.17 - 3.17i)T - 13iT^{2} \)
17 \( 1 + (3.65 + 3.65i)T + 17iT^{2} \)
19 \( 1 + 0.950T + 19T^{2} \)
23 \( 1 + (0.550 + 0.550i)T + 23iT^{2} \)
29 \( 1 + 7.60iT - 29T^{2} \)
31 \( 1 - 7.14iT - 31T^{2} \)
37 \( 1 + (-5.34 + 5.34i)T - 37iT^{2} \)
41 \( 1 + 1.45iT - 41T^{2} \)
43 \( 1 + (-5.98 - 5.98i)T + 43iT^{2} \)
47 \( 1 + (-1.26 - 1.26i)T + 47iT^{2} \)
53 \( 1 + (-7.81 - 7.81i)T + 53iT^{2} \)
59 \( 1 + 4.87T + 59T^{2} \)
61 \( 1 - 9.22iT - 61T^{2} \)
67 \( 1 + (5.94 - 5.94i)T - 67iT^{2} \)
71 \( 1 - 3.91T + 71T^{2} \)
73 \( 1 + (3.60 - 3.60i)T - 73iT^{2} \)
79 \( 1 + 2.27iT - 79T^{2} \)
83 \( 1 + (-2.44 + 2.44i)T - 83iT^{2} \)
89 \( 1 - 8.49T + 89T^{2} \)
97 \( 1 + (-9.19 - 9.19i)T + 97iT^{2} \)
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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

−10.60489715089599955153489139423, −9.366107229307083699347317280828, −8.673570637624571499241282515252, −7.64144965000443823091664000194, −7.12853162809330653879657033721, −6.04943023379510243980592027551, −4.66746107959961082804501593334, −4.15162767364023257083702624245, −2.76258495587644282834166508408, −2.23424329839859488771945730287, 0.63245791762198334427205516473, 1.87091569124683343032652335573, 3.81879479284327693347385458926, 4.60231902817123588812641724920, 5.23530834235665522527243837495, 6.25482405965530736322686398522, 7.27502455258693844391512621383, 7.967775002738330971965699717310, 8.899957618299751740586467321423, 9.828633558950867477426511411777

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