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Properties

Label 2-3150-21.5-c1-0-43
Degree $2$
Conductor $3150$
Sign $0.591 + 0.806i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−2.07 + 1.63i)7-s + 0.999i·8-s + (5.48 − 3.16i)11-s − 1.05i·13-s + (−2.61 + 0.381i)14-s + (−0.5 + 0.866i)16-s + (−2.31 − 4.01i)17-s + (−5.35 − 3.08i)19-s + 6.33·22-s + (−6.10 − 3.52i)23-s + (0.526 − 0.911i)26-s + (−2.45 − 0.978i)28-s − 2.98i·29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.784 + 0.619i)7-s + 0.353i·8-s + (1.65 − 0.954i)11-s − 0.292i·13-s + (−0.699 + 0.101i)14-s + (−0.125 + 0.216i)16-s + (−0.562 − 0.973i)17-s + (−1.22 − 0.708i)19-s + 1.34·22-s + (−1.27 − 0.734i)23-s + (0.103 − 0.178i)26-s + (−0.464 − 0.184i)28-s − 0.553i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.591 + 0.806i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 0.591 + 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.916438746\)
\(L(\frac12)\) \(\approx\) \(1.916438746\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.07 - 1.63i)T \)
good11 \( 1 + (-5.48 + 3.16i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.05iT - 13T^{2} \)
17 \( 1 + (2.31 + 4.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.35 + 3.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.10 + 3.52i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.98iT - 29T^{2} \)
31 \( 1 + (-5.46 + 3.15i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.73 - 3.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.97T + 41T^{2} \)
43 \( 1 - 2.58T + 43T^{2} \)
47 \( 1 + (-4.07 + 7.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.68 - 4.43i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.452 - 0.783i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.81 + 5.08i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.91 + 8.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (-7.24 + 4.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.73 - 4.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + (-1.91 + 3.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.87iT - 97T^{2} \)
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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

−8.664863554961378095653168375203, −7.80144377213962461245729367445, −6.60580144471148875036318335394, −6.41140832835372253905037562120, −5.75527159714879029638913616232, −4.56621646346613519225253550641, −3.98212422720803820399311388714, −3.00129751462183729885817755398, −2.21283320239505910091933237073, −0.47271760478886110235992488195, 1.35559908263530073749423032228, 2.14200144019716601291644678416, 3.50586671615298455417087531868, 4.10399183843965964735666158572, 4.53571673190782409453843066575, 6.08379195527062105025416436385, 6.30334785067475623247071151560, 7.06988355124435227967042362559, 7.957252347365337367385029721279, 9.062758457218873953545435412891

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