Properties

Label 2-3150-105.89-c1-0-8
Degree $2$
Conductor $3150$
Sign $0.0156 - 0.999i$
Analytic cond. $25.1528$
Root an. cond. $5.01526$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.22 + 2.34i)7-s − 0.999·8-s + (−2.03 + 1.17i)11-s + 4.64·13-s + (2.64 + 0.107i)14-s + (−0.5 + 0.866i)16-s + (−3.95 + 2.28i)17-s + (−0.491 − 0.283i)19-s + 2.35i·22-s + (−2.91 + 5.04i)23-s + (2.32 − 4.02i)26-s + (1.41 − 2.23i)28-s + 2.55i·29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.464 + 0.885i)7-s − 0.353·8-s + (−0.615 + 0.355i)11-s + 1.28·13-s + (0.706 + 0.0287i)14-s + (−0.125 + 0.216i)16-s + (−0.958 + 0.553i)17-s + (−0.112 − 0.0650i)19-s + 0.502i·22-s + (−0.607 + 1.05i)23-s + (0.455 − 0.789i)26-s + (0.267 − 0.422i)28-s + 0.474i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0156 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0156 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.0156 - 0.999i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 0.0156 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.156894782\)
\(L(\frac12)\) \(\approx\) \(1.156894782\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-1.22 - 2.34i)T \)
good11 \( 1 + (2.03 - 1.17i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.64T + 13T^{2} \)
17 \( 1 + (3.95 - 2.28i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.491 + 0.283i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.91 - 5.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.55iT - 29T^{2} \)
31 \( 1 + (1.89 - 1.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.02 + 4.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.68T + 41T^{2} \)
43 \( 1 + 6.57iT - 43T^{2} \)
47 \( 1 + (5.46 + 3.15i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.07 - 10.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.67 - 2.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.85 + 3.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.46 + 2.00i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.02iT - 71T^{2} \)
73 \( 1 + (-4.10 - 7.11i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.13 - 7.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.171iT - 83T^{2} \)
89 \( 1 + (2.72 - 4.72i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−8.683673066068053302598271830673, −8.512429487444482529987242523702, −7.37497092806607697850525863408, −6.44027424956792225480947600337, −5.61296087763825294912199692706, −5.10689455519628062718134507076, −4.05996110117545032117562270388, −3.33972648364844027007901895654, −2.18626630238888384529277148676, −1.56579600883424547065173667266, 0.29806951271406331045874005760, 1.75840455364425734995649673574, 3.06562652647156814515968014621, 3.93346844291212078676549597930, 4.65383313823516150257516074289, 5.39818509631213904422033834000, 6.43666904042455568540333323342, 6.78411146389877017722614185450, 7.84771499461703230869693180226, 8.315176696565796243202814622380

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