Properties

Label 2-15-5.2-c4-0-0
Degree $2$
Conductor $15$
Sign $0.477 - 0.878i$
Analytic cond. $1.55054$
Root an. cond. $1.24521$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.80 + 3.80i)2-s + (−3.67 + 3.67i)3-s + 12.9i·4-s + (6.56 − 24.1i)5-s − 27.9·6-s + (16.6 + 16.6i)7-s + (11.6 − 11.6i)8-s − 27i·9-s + (116. − 66.7i)10-s − 215.·11-s + (−47.5 − 47.5i)12-s + (−29.9 + 29.9i)13-s + 126. i·14-s + (64.4 + 112. i)15-s + 295.·16-s + (−3.97 − 3.97i)17-s + ⋯
L(s)  = 1  + (0.950 + 0.950i)2-s + (−0.408 + 0.408i)3-s + 0.808i·4-s + (0.262 − 0.964i)5-s − 0.776·6-s + (0.339 + 0.339i)7-s + (0.182 − 0.182i)8-s − 0.333i·9-s + (1.16 − 0.667i)10-s − 1.77·11-s + (−0.329 − 0.329i)12-s + (−0.177 + 0.177i)13-s + 0.644i·14-s + (0.286 + 0.501i)15-s + 1.15·16-s + (−0.0137 − 0.0137i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(1.55054\)
Root analytic conductor: \(1.24521\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :2),\ 0.477 - 0.878i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.38748 + 0.825193i\)
\(L(\frac12)\) \(\approx\) \(1.38748 + 0.825193i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.67 - 3.67i)T \)
5 \( 1 + (-6.56 + 24.1i)T \)
good2 \( 1 + (-3.80 - 3.80i)T + 16iT^{2} \)
7 \( 1 + (-16.6 - 16.6i)T + 2.40e3iT^{2} \)
11 \( 1 + 215.T + 1.46e4T^{2} \)
13 \( 1 + (29.9 - 29.9i)T - 2.85e4iT^{2} \)
17 \( 1 + (3.97 + 3.97i)T + 8.35e4iT^{2} \)
19 \( 1 - 604. iT - 1.30e5T^{2} \)
23 \( 1 + (-376. + 376. i)T - 2.79e5iT^{2} \)
29 \( 1 + 624. iT - 7.07e5T^{2} \)
31 \( 1 - 263.T + 9.23e5T^{2} \)
37 \( 1 + (-979. - 979. i)T + 1.87e6iT^{2} \)
41 \( 1 + 1.57e3T + 2.82e6T^{2} \)
43 \( 1 + (30.8 - 30.8i)T - 3.41e6iT^{2} \)
47 \( 1 + (-1.78e3 - 1.78e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-707. + 707. i)T - 7.89e6iT^{2} \)
59 \( 1 - 1.36e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.98e3T + 1.38e7T^{2} \)
67 \( 1 + (986. + 986. i)T + 2.01e7iT^{2} \)
71 \( 1 - 68.0T + 2.54e7T^{2} \)
73 \( 1 + (-2.47e3 + 2.47e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 6.04e3iT - 3.89e7T^{2} \)
83 \( 1 + (5.15e3 - 5.15e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 7.02e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.08e4 - 1.08e4i)T + 8.85e7iT^{2} \)
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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

−18.56905541616825762887809168084, −16.93471579995188539587622865357, −16.04794905751432022021878884606, −14.99369749455235673972033937233, −13.46315439147542628988227884842, −12.34007233433902482612540196255, −10.19231231875851234682738678634, −8.050361921473410093385612610457, −5.78953451208679608495895984839, −4.74065633232060740198419234656, 2.69053713140465725591469529110, 5.18962143882425666349782604269, 7.43025333046922290462064908300, 10.51568288832202421767913082244, 11.28188727905201959736316894457, 12.94604152262597915754495913626, 13.77583484826141730434978393703, 15.32525655902359597836617622054, 17.40999244145408431496035758779, 18.51611308588517442290462922874

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