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Properties

Label 2-945-315.157-c1-0-8
Degree $2$
Conductor $945$
Sign $-0.217 - 0.975i$
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  + (−2.03 + 0.544i)2-s + (2.10 − 1.21i)4-s + (−2.13 − 0.656i)5-s + (2.62 + 0.310i)7-s + (−0.638 + 0.638i)8-s + (4.70 + 0.170i)10-s + 0.0883·11-s + (−5.14 + 1.37i)13-s + (−5.50 + 0.800i)14-s + (−1.47 + 2.56i)16-s + (0.913 + 3.40i)17-s + (−2.57 − 4.46i)19-s + (−5.29 + 1.21i)20-s + (−0.179 + 0.0481i)22-s + (2.62 − 2.62i)23-s + ⋯
L(s)  = 1  + (−1.43 + 0.385i)2-s + (1.05 − 0.607i)4-s + (−0.955 − 0.293i)5-s + (0.993 + 0.117i)7-s + (−0.225 + 0.225i)8-s + (1.48 + 0.0538i)10-s + 0.0266·11-s + (−1.42 + 0.382i)13-s + (−1.47 + 0.214i)14-s + (−0.369 + 0.640i)16-s + (0.221 + 0.826i)17-s + (−0.591 − 1.02i)19-s + (−1.18 + 0.271i)20-s + (−0.0382 + 0.0102i)22-s + (0.546 − 0.546i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.217 - 0.975i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.217 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278050 + 0.346995i\)
\(L(\frac12)\) \(\approx\) \(0.278050 + 0.346995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 + 0.656i)T \)
7 \( 1 + (-2.62 - 0.310i)T \)
good2 \( 1 + (2.03 - 0.544i)T + (1.73 - i)T^{2} \)
11 \( 1 - 0.0883T + 11T^{2} \)
13 \( 1 + (5.14 - 1.37i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.913 - 3.40i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.57 + 4.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.62 + 2.62i)T - 23iT^{2} \)
29 \( 1 + (-0.609 + 0.351i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.22 + 2.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.26 - 8.44i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.57 + 1.48i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.22 - 0.595i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.48 - 5.54i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.63 - 13.5i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.73 + 2.99i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.18 + 0.682i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.54 - 5.78i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + (1.06 - 0.285i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.07 - 1.77i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0856 - 0.319i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-5.71 - 9.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.0 + 2.70i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.22675561066291083876882322540, −9.128041537592328190228766842564, −8.627150222739251007120827227747, −7.84792365782851187458463983084, −7.32928400950671014912598935971, −6.42253451050576242813503840062, −4.88688584335544692271490960167, −4.29355826790205407925407615868, −2.50090914015381173557397090314, −1.09588223309811962981172467481, 0.40399778315511047678288058745, 1.88934329366121961968681615003, 3.04474787482863863954937649912, 4.44439619544865330951742740347, 5.32262846495852543933216092743, 7.07759758117646567824472149119, 7.48327035612822086175103230034, 8.210184499030544917194579815417, 8.874209635234663418879527563795, 9.946753408219954547501087639983

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