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Properties

Label 2-945-105.89-c1-0-40
Degree $2$
Conductor $945$
Sign $0.532 - 0.846i$
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 + (2.23 + 0.135i)5-s + (2.46 + 0.967i)7-s − 1.44·8-s + (−0.461 + 0.697i)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 + (0.998 + 0.0603i)5-s + (0.930 + 0.365i)7-s − 0.510·8-s + (−0.145 + 0.220i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.532 - 0.846i$
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: \((2,\ 945,\ (\ :1/2),\ 0.532 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95862 + 1.08162i\)
\(L(\frac12)\) \(\approx\) \(1.95862 + 1.08162i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.135i)T \)
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

−10.28022369756261148695770227801, −8.904394112498252894349753751400, −8.690242157374881199578443156224, −7.78232004146650652934376800082, −6.55086497493573815629969992964, −6.18608881833532846947954843165, −5.00432794824054761976023226719, −3.83568369229194799345263585680, −2.61589372274580626701433103311, −1.61710405086483995014112861661, 1.35095333833539025533124222218, 1.87798499786205880969646864751, 3.43350292199095634704780982606, 4.78987467482613937983137116962, 5.65738031742259057570242471417, 6.36601176708467463183956061741, 7.21588133524581915071390526252, 8.537496374256799790799933638366, 9.071387633281809105465727879492, 10.12694696374139142782383140593

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