↑

Properties

Label 2-945-15.2-c1-0-38
Degree $2$
Conductor $945$
Sign $0.329 + 0.944i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.49 − 1.49i)2-s − 2.48i·4-s + (2.18 + 0.482i)5-s + (0.707 + 0.707i)7-s + (−0.732 − 0.732i)8-s + (3.99 − 2.54i)10-s − 0.466i·11-s + (−0.380 + 0.380i)13-s + 2.11·14-s + 2.78·16-s + (−0.265 + 0.265i)17-s − 4.44i·19-s + (1.20 − 5.43i)20-s + (−0.699 − 0.699i)22-s + (1.85 + 1.85i)23-s + ⋯
L(s)  = 1  + (1.05 − 1.05i)2-s − 1.24i·4-s + (0.976 + 0.215i)5-s + (0.267 + 0.267i)7-s + (−0.258 − 0.258i)8-s + (1.26 − 0.805i)10-s − 0.140i·11-s + (−0.105 + 0.105i)13-s + 0.566·14-s + 0.695·16-s + (−0.0643 + 0.0643i)17-s − 1.01i·19-s + (0.268 − 1.21i)20-s + (−0.149 − 0.149i)22-s + (0.387 + 0.387i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.329 + 0.944i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.78424 - 1.97654i\)
\(L(\frac12)\) \(\approx\) \(2.78424 - 1.97654i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 0.482i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-1.49 + 1.49i)T - 2iT^{2} \)
11 \( 1 + 0.466iT - 11T^{2} \)
13 \( 1 + (0.380 - 0.380i)T - 13iT^{2} \)
17 \( 1 + (0.265 - 0.265i)T - 17iT^{2} \)
19 \( 1 + 4.44iT - 19T^{2} \)
23 \( 1 + (-1.85 - 1.85i)T + 23iT^{2} \)
29 \( 1 + 5.05T + 29T^{2} \)
31 \( 1 - 4.03T + 31T^{2} \)
37 \( 1 + (-0.795 - 0.795i)T + 37iT^{2} \)
41 \( 1 + 5.98iT - 41T^{2} \)
43 \( 1 + (7.10 - 7.10i)T - 43iT^{2} \)
47 \( 1 + (-2.04 + 2.04i)T - 47iT^{2} \)
53 \( 1 + (8.40 + 8.40i)T + 53iT^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 + 9.69T + 61T^{2} \)
67 \( 1 + (-8.98 - 8.98i)T + 67iT^{2} \)
71 \( 1 - 5.63iT - 71T^{2} \)
73 \( 1 + (-1.01 + 1.01i)T - 73iT^{2} \)
79 \( 1 - 13.3iT - 79T^{2} \)
83 \( 1 + (11.7 + 11.7i)T + 83iT^{2} \)
89 \( 1 + 2.85T + 89T^{2} \)
97 \( 1 + (5.27 + 5.27i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.04926295624187530591777253640, −9.396431086820353676076923280320, −8.375309955735043350361702128924, −7.12744025328926899405585295680, −6.09700904292199164674965641672, −5.27302728771131277452111860849, −4.57212188170952753878831977522, −3.30784702608378299293857846403, −2.47694849239863748788623028061, −1.49540144851673710021192452289, 1.58538344666775815905036754329, 3.12658626007434807579513956488, 4.34706520676063358147081506457, 5.06239405064703325754425941820, 5.91109471577688274470978083827, 6.52011494796256496280336100996, 7.48366292419610837265235097113, 8.242417856325577415815365326966, 9.346653024553905543462131534388, 10.15422286876581126949942821705

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