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Properties

Label 2-15-15.2-c17-0-19
Degree $2$
Conductor $15$
Sign $0.934 + 0.356i$
Analytic cond. $27.4833$
Root an. cond. $5.24245$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (431. − 431. i)2-s + (9.57e3 + 6.12e3i)3-s − 2.41e5i·4-s + (−2.98e5 + 8.20e5i)5-s + (6.77e6 − 1.48e6i)6-s + (9.48e6 + 9.48e6i)7-s + (−4.77e7 − 4.77e7i)8-s + (5.40e7 + 1.17e8i)9-s + (2.25e8 + 4.83e8i)10-s + 3.89e8i·11-s + (1.47e9 − 2.31e9i)12-s + (3.08e9 − 3.08e9i)13-s + 8.19e9·14-s + (−7.88e9 + 6.02e9i)15-s − 9.52e9·16-s + (1.31e10 − 1.31e10i)17-s + ⋯
L(s)  = 1  + (1.19 − 1.19i)2-s + (0.842 + 0.539i)3-s − 1.84i·4-s + (−0.341 + 0.939i)5-s + (1.64 − 0.361i)6-s + (0.622 + 0.622i)7-s + (−1.00 − 1.00i)8-s + (0.418 + 0.908i)9-s + (0.713 + 1.52i)10-s + 0.547i·11-s + (0.993 − 1.55i)12-s + (1.04 − 1.04i)13-s + 1.48·14-s + (−0.794 + 0.607i)15-s − 0.554·16-s + (0.457 − 0.457i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.934 + 0.356i$
Analytic conductor: \(27.4833\)
Root analytic conductor: \(5.24245\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :17/2),\ 0.934 + 0.356i)\)

Particular Values

\(L(9)\) \(\approx\) \(5.249570501\)
\(L(\frac12)\) \(\approx\) \(5.249570501\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-9.57e3 - 6.12e3i)T \)
5 \( 1 + (2.98e5 - 8.20e5i)T \)
good2 \( 1 + (-431. + 431. i)T - 1.31e5iT^{2} \)
7 \( 1 + (-9.48e6 - 9.48e6i)T + 2.32e14iT^{2} \)
11 \( 1 - 3.89e8iT - 5.05e17T^{2} \)
13 \( 1 + (-3.08e9 + 3.08e9i)T - 8.65e18iT^{2} \)
17 \( 1 + (-1.31e10 + 1.31e10i)T - 8.27e20iT^{2} \)
19 \( 1 - 1.06e11iT - 5.48e21T^{2} \)
23 \( 1 + (3.27e10 + 3.27e10i)T + 1.41e23iT^{2} \)
29 \( 1 + 2.84e12T + 7.25e24T^{2} \)
31 \( 1 - 7.19e12T + 2.25e25T^{2} \)
37 \( 1 + (2.24e13 + 2.24e13i)T + 4.56e26iT^{2} \)
41 \( 1 + 3.01e13iT - 2.61e27T^{2} \)
43 \( 1 + (-1.73e13 + 1.73e13i)T - 5.87e27iT^{2} \)
47 \( 1 + (1.43e14 - 1.43e14i)T - 2.66e28iT^{2} \)
53 \( 1 + (-6.60e12 - 6.60e12i)T + 2.05e29iT^{2} \)
59 \( 1 + 3.73e14T + 1.27e30T^{2} \)
61 \( 1 - 2.96e14T + 2.24e30T^{2} \)
67 \( 1 + (3.31e15 + 3.31e15i)T + 1.10e31iT^{2} \)
71 \( 1 - 8.74e15iT - 2.96e31T^{2} \)
73 \( 1 + (-4.49e15 + 4.49e15i)T - 4.74e31iT^{2} \)
79 \( 1 - 1.83e16iT - 1.81e32T^{2} \)
83 \( 1 + (-8.95e15 - 8.95e15i)T + 4.21e32iT^{2} \)
89 \( 1 - 2.51e16T + 1.37e33T^{2} \)
97 \( 1 + (8.44e16 + 8.44e16i)T + 5.95e33iT^{2} \)
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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

−14.71986967366013179989004976487, −13.85310807710901718795378284636, −12.35040498305085049587527456866, −11.01635830249834558336564074568, −10.05257611383197669759023863938, −8.012216406071266835259824833795, −5.52951765133199355825811210608, −3.95357255096768444335680334244, −2.99808511746945751888227714820, −1.79306268542200749055919645461, 1.23065841874960724557340730422, 3.60121564811002190114240324480, 4.68649788834966040556829450953, 6.46016319848369141947499024240, 7.79300490965853688810931301256, 8.765746738084320961053454248516, 11.75951252943494105090474691742, 13.29238160062222559337571873410, 13.75951907671235777911160044127, 15.07986662826152902271615255112

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