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Properties

Label 2-945-315.283-c1-0-23
Degree $2$
Conductor $945$
Sign $0.973 - 0.226i$
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  + (0.0302 + 0.112i)2-s + (1.72 − 0.993i)4-s + (0.634 + 2.14i)5-s + (2.46 + 0.960i)7-s + (0.329 + 0.329i)8-s + (−0.222 + 0.136i)10-s + 0.911·11-s + (−1.53 − 5.74i)13-s + (−0.0338 + 0.307i)14-s + (1.95 − 3.39i)16-s + (3.29 − 0.884i)17-s + (0.815 + 1.41i)19-s + (3.22 + 3.05i)20-s + (0.0275 + 0.102i)22-s + (2.30 + 2.30i)23-s + ⋯
L(s)  = 1  + (0.0213 + 0.0797i)2-s + (0.860 − 0.496i)4-s + (0.283 + 0.958i)5-s + (0.931 + 0.363i)7-s + (0.116 + 0.116i)8-s + (−0.0704 + 0.0431i)10-s + 0.274·11-s + (−0.426 − 1.59i)13-s + (−0.00905 + 0.0820i)14-s + (0.489 − 0.848i)16-s + (0.800 − 0.214i)17-s + (0.187 + 0.323i)19-s + (0.720 + 0.683i)20-s + (0.00587 + 0.0219i)22-s + (0.481 + 0.481i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.973 - 0.226i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.973 - 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31991 + 0.266695i\)
\(L(\frac12)\) \(\approx\) \(2.31991 + 0.266695i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.634 - 2.14i)T \)
7 \( 1 + (-2.46 - 0.960i)T \)
good2 \( 1 + (-0.0302 - 0.112i)T + (-1.73 + i)T^{2} \)
11 \( 1 - 0.911T + 11T^{2} \)
13 \( 1 + (1.53 + 5.74i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-3.29 + 0.884i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.815 - 1.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.30 - 2.30i)T + 23iT^{2} \)
29 \( 1 + (5.74 - 3.31i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.00 - 0.581i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.49 - 0.935i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (5.95 + 3.43i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.11 - 4.14i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (5.64 - 1.51i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-13.5 + 3.64i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.06 + 3.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.22 + 3.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-14.7 - 3.94i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.48T + 71T^{2} \)
73 \( 1 + (-0.993 - 3.70i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.91 - 4.57i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.7 + 2.88i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (1.92 + 3.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.83 - 6.83i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.21058845708911544149375861393, −9.506066490818668234093533069505, −8.104755881203417671477180707102, −7.52034921055453541260694092723, −6.72729592918506086945944611191, −5.54508532014538977829990313283, −5.33028692272616266679045502698, −3.45146265359418002871863026338, −2.57697279753921462164180648641, −1.42682493696585572059222753087, 1.38561916309489776386371459420, 2.24163459148783086473759618133, 3.83368559661682691452343311944, 4.61675413626539253106818781781, 5.64382589980136789814704272207, 6.75032850705387064694018560088, 7.49688029874334754379017057517, 8.320779986661621454457280580059, 9.111961623661091032456335783999, 9.994923859714221472188048782336

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