Properties

Label 2-945-945.734-c1-0-123
Degree $2$
Conductor $945$
Sign $0.240 + 0.970i$
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  + (1.62 + 0.586i)3-s + (−0.347 + 1.96i)4-s + (−0.764 − 2.10i)5-s + (−0.459 − 2.60i)7-s + (2.31 + 1.91i)9-s + (2.20 − 6.07i)11-s + (−1.72 + 3.00i)12-s + (−3.66 − 3.07i)13-s + (−0.0137 − 3.87i)15-s + (−3.75 − 1.36i)16-s + (−6.43 − 3.71i)17-s + (4.40 − 0.776i)20-s + (0.779 − 4.51i)21-s + (−3.83 + 3.21i)25-s + (2.64 + 4.47i)27-s + 5.29·28-s + ⋯
L(s)  = 1  + (0.940 + 0.338i)3-s + (−0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s + (−0.173 − 0.984i)7-s + (0.770 + 0.637i)9-s + (0.666 − 1.83i)11-s + (−0.496 + 0.867i)12-s + (−1.01 − 0.852i)13-s + (−0.00354 − 0.999i)15-s + (−0.939 − 0.342i)16-s + (−1.56 − 0.900i)17-s + (0.984 − 0.173i)20-s + (0.170 − 0.985i)21-s + (−0.766 + 0.642i)25-s + (0.509 + 0.860i)27-s + 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.240 + 0.970i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.240 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22025 - 0.954356i\)
\(L(\frac12)\) \(\approx\) \(1.22025 - 0.954356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 - 0.586i)T \)
5 \( 1 + (0.764 + 2.10i)T \)
7 \( 1 + (0.459 + 2.60i)T \)
good2 \( 1 + (0.347 - 1.96i)T^{2} \)
11 \( 1 + (-2.20 + 6.07i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (3.66 + 3.07i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (6.43 + 3.71i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-3.10 - 3.70i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-13.0 + 2.29i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-12.0 - 6.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.68 - 9.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.15 + 6.84i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.15 - 2.56i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.22 + 3.35i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.541087822528184304430898445269, −8.866897393226400721294284317389, −8.357103464645626055140458369548, −7.57199476160779041525216261353, −6.80628020458045071978220156687, −5.14168611434839543268547262391, −4.26761044509113995917967072238, −3.58206084741453424397585170447, −2.66346118819429920313660608297, −0.62599268018819144891395713597, 2.09155794312357655235887297366, 2.27174985008607361999612047431, 4.04513812182541576505533194066, 4.71550024212255112044410233917, 6.35557003815050612547419524922, 6.71269237030343510271876529939, 7.57256564143212175824365072971, 8.786559651436862044426042868623, 9.439906989767752764434538804454, 9.935474536962757713009252948259

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