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Properties

Label 2-945-105.89-c1-0-7
Degree $2$
Conductor $945$
Sign $-0.996 + 0.0826i$
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  + (−0.187 + 0.323i)2-s + (0.930 + 1.61i)4-s + (1.23 + 1.86i)5-s + (−2.46 − 0.967i)7-s − 1.44·8-s + (−0.834 + 0.0505i)10-s + (−0.947 + 0.547i)11-s − 5.52·13-s + (0.774 − 0.616i)14-s + (−1.58 + 2.75i)16-s + (−0.343 + 0.198i)17-s + (−4.00 − 2.31i)19-s + (−1.85 + 3.72i)20-s − 0.409i·22-s + (1.81 − 3.13i)23-s + ⋯
L(s)  = 1  + (−0.132 + 0.229i)2-s + (0.465 + 0.805i)4-s + (0.551 + 0.834i)5-s + (−0.930 − 0.365i)7-s − 0.510·8-s + (−0.264 + 0.0159i)10-s + (−0.285 + 0.165i)11-s − 1.53·13-s + (0.206 − 0.164i)14-s + (−0.397 + 0.688i)16-s + (−0.0833 + 0.0481i)17-s + (−0.919 − 0.531i)19-s + (−0.415 + 0.832i)20-s − 0.0872i·22-s + (0.377 − 0.654i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.996 + 0.0826i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.996 + 0.0826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0301661 - 0.728596i\)
\(L(\frac12)\) \(\approx\) \(0.0301661 - 0.728596i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 1.86i)T \)
7 \( 1 + (2.46 + 0.967i)T \)
good2 \( 1 + (0.187 - 0.323i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (0.947 - 0.547i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.52T + 13T^{2} \)
17 \( 1 + (0.343 - 0.198i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.00 + 2.31i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.81 + 3.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.19iT - 29T^{2} \)
31 \( 1 + (3.46 - 1.99i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.338 + 0.195i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 3.44iT - 43T^{2} \)
47 \( 1 + (4.21 + 2.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.58 - 4.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.11 - 7.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.91 + 1.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.20 + 2.42i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.84iT - 71T^{2} \)
73 \( 1 + (-0.357 - 0.618i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.40 - 4.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.9iT - 83T^{2} \)
89 \( 1 + (6.38 - 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.84T + 97T^{2} \)
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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

−10.55526768583273620245623976492, −9.605999943755508281630544484898, −8.903439844006417589923739814959, −7.68957219444406230423942168160, −6.93815948502009671234283099787, −6.64474227253500964510311475517, −5.42205562119487668143439591479, −4.09052674823856649305834588157, −2.92999862519955665582577409685, −2.36156562843371720279880102169, 0.31292442138320657644995414092, 1.95323013839382985557008779514, 2.76365916243273974558977713847, 4.38191628117126483683753497860, 5.46173634905257006669758607700, 5.98407516160511552930132639206, 6.92539159980637043770256733859, 8.010594466902753445195604004725, 9.172972715921110727230967905201, 9.671101898082235950209876087496

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