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Properties

Label 2-15-5.2-c2-0-1
Degree $2$
Conductor $15$
Sign $0.326 + 0.945i$
Analytic cond. $0.408720$
Root an. cond. $0.639312$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.22 − 2.22i)2-s + (1.22 − 1.22i)3-s + 5.89i·4-s + (2.67 + 4.22i)5-s − 5.44·6-s + (−1.44 − 1.44i)7-s + (4.22 − 4.22i)8-s − 2.99i·9-s + (3.44 − 15.3i)10-s − 3.34·11-s + (7.22 + 7.22i)12-s + (−10.4 + 10.4i)13-s + 6.44i·14-s + (8.44 + 1.89i)15-s + 4.79·16-s + (−2.65 − 2.65i)17-s + ⋯
L(s)  = 1  + (−1.11 − 1.11i)2-s + (0.408 − 0.408i)3-s + 1.47i·4-s + (0.534 + 0.844i)5-s − 0.908·6-s + (−0.207 − 0.207i)7-s + (0.528 − 0.528i)8-s − 0.333i·9-s + (0.344 − 1.53i)10-s − 0.304·11-s + (0.602 + 0.602i)12-s + (−0.803 + 0.803i)13-s + 0.460i·14-s + (0.563 + 0.126i)15-s + 0.299·16-s + (−0.155 − 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.326 + 0.945i$
Analytic conductor: \(0.408720\)
Root analytic conductor: \(0.639312\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :1),\ 0.326 + 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.452246 - 0.322280i\)
\(L(\frac12)\) \(\approx\) \(0.452246 - 0.322280i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (-2.67 - 4.22i)T \)
good2 \( 1 + (2.22 + 2.22i)T + 4iT^{2} \)
7 \( 1 + (1.44 + 1.44i)T + 49iT^{2} \)
11 \( 1 + 3.34T + 121T^{2} \)
13 \( 1 + (10.4 - 10.4i)T - 169iT^{2} \)
17 \( 1 + (2.65 + 2.65i)T + 289iT^{2} \)
19 \( 1 + 20.6iT - 361T^{2} \)
23 \( 1 + (-16.4 + 16.4i)T - 529iT^{2} \)
29 \( 1 - 0.853iT - 841T^{2} \)
31 \( 1 + 18.6T + 961T^{2} \)
37 \( 1 + (-38.0 - 38.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 28.6T + 1.68e3T^{2} \)
43 \( 1 + (-22.4 + 22.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-19.7 - 19.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-28.6 + 28.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 111. iT - 3.48e3T^{2} \)
61 \( 1 - 94.0T + 3.72e3T^{2} \)
67 \( 1 + (54.8 + 54.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 68T + 5.04e3T^{2} \)
73 \( 1 + (39.7 - 39.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (21.1 - 21.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 94.1iT - 7.92e3T^{2} \)
97 \( 1 + (-14.5 - 14.5i)T + 9.40e3iT^{2} \)
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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

−18.96903735726170195732808359998, −18.12221688732551110378013755514, −16.98358085609735254137258722380, −14.75812454433060953282430304298, −13.21971398269982858515992813489, −11.57135144001210906439504254971, −10.24451000268981907618260024464, −9.033008024282958162235044981256, −7.11450047940662188936748839910, −2.58030800527451952960517598202, 5.59617306224645946076413331450, 7.77350152759057367301298746720, 9.082219061602929262321369428557, 10.11337379201305347934855388501, 12.80708065831191322667351679521, 14.63331949910544798040443994802, 15.82365343681964090392548396174, 16.82410332212201973333963080470, 17.82451588551342425590115748233, 19.22679516271754697526217925726

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