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Properties

Label 2-945-315.223-c1-0-41
Degree $2$
Conductor $945$
Sign $-0.999 - 0.0284i$
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  + (−0.388 − 1.44i)2-s + (−0.218 + 0.126i)4-s + (1.66 − 1.48i)5-s + (2.18 − 1.49i)7-s + (−1.85 − 1.85i)8-s + (−2.80 − 1.84i)10-s + (−1.50 + 2.60i)11-s + (−5.41 − 1.45i)13-s + (−3.01 − 2.58i)14-s + (−2.22 + 3.84i)16-s + (−1.04 − 1.04i)17-s − 7.21·19-s + (−0.177 + 0.536i)20-s + (4.36 + 1.16i)22-s + (1.46 − 5.45i)23-s + ⋯
L(s)  = 1  + (−0.274 − 1.02i)2-s + (−0.109 + 0.0631i)4-s + (0.746 − 0.665i)5-s + (0.824 − 0.565i)7-s + (−0.655 − 0.655i)8-s + (−0.887 − 0.582i)10-s + (−0.453 + 0.786i)11-s + (−1.50 − 0.402i)13-s + (−0.805 − 0.690i)14-s + (−0.555 + 0.961i)16-s + (−0.254 − 0.254i)17-s − 1.65·19-s + (−0.0396 + 0.119i)20-s + (0.930 + 0.249i)22-s + (0.304 − 1.13i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.999 - 0.0284i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.999 - 0.0284i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0191117 + 1.34275i\)
\(L(\frac12)\) \(\approx\) \(0.0191117 + 1.34275i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 + 1.48i)T \)
7 \( 1 + (-2.18 + 1.49i)T \)
good2 \( 1 + (0.388 + 1.44i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (1.50 - 2.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.41 + 1.45i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.04 + 1.04i)T + 17iT^{2} \)
19 \( 1 + 7.21T + 19T^{2} \)
23 \( 1 + (-1.46 + 5.45i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.88 - 2.82i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.98 + 2.87i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.33 + 2.33i)T - 37iT^{2} \)
41 \( 1 + (-3.05 + 1.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.96 + 0.793i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (5.88 - 1.57i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.84 - 6.84i)T + 53iT^{2} \)
59 \( 1 + (2.39 + 4.13i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.10 + 2.94i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.23 - 0.865i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.34T + 71T^{2} \)
73 \( 1 + (-3.61 + 3.61i)T - 73iT^{2} \)
79 \( 1 + (8.69 + 5.01i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.149 - 0.559i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 0.259T + 89T^{2} \)
97 \( 1 + (-12.3 + 3.32i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.940369877217394615307370720122, −9.023789011091212598157840988119, −8.168239918228456603685503414629, −7.11603819337341419569287404733, −6.18695708451154214403168984750, −4.84372578461328652808786417007, −4.42856550954704196899292018205, −2.56899469202100272091830979846, −2.05003807414968075038844515846, −0.63283888025035996212526762936, 2.12946366074385809046529548367, 2.83510798832277108441178394964, 4.66650293646240297386427849110, 5.53701154904841129214286694157, 6.29525968106510979740742608947, 7.01453355101647210869697311242, 7.954472245126385266228508842010, 8.560020707129551654425934085648, 9.449970573500924137283512558029, 10.37728793816822869578735980518

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