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Properties

Label 2-945-105.104-c1-0-16
Degree $2$
Conductor $945$
Sign $0.526 - 0.850i$
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  + 0.631·2-s − 1.60·4-s + (−2.21 − 0.276i)5-s + (2.40 + 1.10i)7-s − 2.27·8-s + (−1.40 − 0.174i)10-s − 4.53i·11-s + 0.148·13-s + (1.51 + 0.697i)14-s + 1.76·16-s + 5.50i·17-s + 4.84i·19-s + (3.55 + 0.442i)20-s − 2.86i·22-s + 5.63·23-s + ⋯
L(s)  = 1  + 0.446·2-s − 0.800·4-s + (−0.992 − 0.123i)5-s + (0.908 + 0.417i)7-s − 0.803·8-s + (−0.443 − 0.0552i)10-s − 1.36i·11-s + 0.0411·13-s + (0.405 + 0.186i)14-s + 0.441·16-s + 1.33i·17-s + 1.11i·19-s + (0.794 + 0.0990i)20-s − 0.609i·22-s + 1.17·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.526 - 0.850i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (944, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.526 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08242 + 0.602683i\)
\(L(\frac12)\) \(\approx\) \(1.08242 + 0.602683i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 0.276i)T \)
7 \( 1 + (-2.40 - 1.10i)T \)
good2 \( 1 - 0.631T + 2T^{2} \)
11 \( 1 + 4.53iT - 11T^{2} \)
13 \( 1 - 0.148T + 13T^{2} \)
17 \( 1 - 5.50iT - 17T^{2} \)
19 \( 1 - 4.84iT - 19T^{2} \)
23 \( 1 - 5.63T + 23T^{2} \)
29 \( 1 - 5.60iT - 29T^{2} \)
31 \( 1 - 9.41iT - 31T^{2} \)
37 \( 1 - 5.50iT - 37T^{2} \)
41 \( 1 + 3.55T + 41T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 + 3.21iT - 47T^{2} \)
53 \( 1 + 8.99T + 53T^{2} \)
59 \( 1 - 7.28T + 59T^{2} \)
61 \( 1 - 8.15iT - 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + 5.66iT - 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 0.0389iT - 83T^{2} \)
89 \( 1 - 8.78T + 89T^{2} \)
97 \( 1 - 4.42T + 97T^{2} \)
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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

−10.44080516640862470045284557638, −8.906702269201522834276641085777, −8.532396767404043542804433502408, −8.014725706689100982406152405859, −6.68880092321232282076033869124, −5.52471010707919541211498336593, −4.97868278358262487700138038018, −3.82504985942026018567751451417, −3.25400370693628414207387452316, −1.22518317815910003338918203180, 0.61688423997280492200266771527, 2.57001423020414790190444655226, 3.84960734967018854530708463958, 4.75683792018343328330726348528, 4.94819078967861962351288336755, 6.60616355813903935041293188878, 7.53818554382288622864956190297, 8.019412240311792932886806860093, 9.226750050911592876914766558667, 9.679461820313705005023561519575

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