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Properties

Label 2-15-15.2-c17-0-27
Degree $2$
Conductor $15$
Sign $-0.852 + 0.522i$
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  + (−244. + 244. i)2-s + (9.39e3 − 6.38e3i)3-s + 1.17e4i·4-s + (4.94e5 − 7.20e5i)5-s + (−7.34e5 + 3.85e6i)6-s + (−5.02e6 − 5.02e6i)7-s + (−3.48e7 − 3.48e7i)8-s + (4.74e7 − 1.20e8i)9-s + (5.51e7 + 2.96e8i)10-s − 8.18e8i·11-s + (7.51e7 + 1.10e8i)12-s + (−3.71e9 + 3.71e9i)13-s + 2.45e9·14-s + (4.29e7 − 9.92e9i)15-s + 1.55e10·16-s + (−1.47e10 + 1.47e10i)17-s + ⋯
L(s)  = 1  + (−0.674 + 0.674i)2-s + (0.826 − 0.562i)3-s + 0.0897i·4-s + (0.565 − 0.824i)5-s + (−0.178 + 0.937i)6-s + (−0.329 − 0.329i)7-s + (−0.735 − 0.735i)8-s + (0.367 − 0.929i)9-s + (0.174 + 0.937i)10-s − 1.15i·11-s + (0.0504 + 0.0741i)12-s + (−1.26 + 1.26i)13-s + 0.444·14-s + (0.00432 − 0.999i)15-s + 0.902·16-s + (−0.513 + 0.513i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.852 + 0.522i$
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: \((2,\ 15,\ (\ :17/2),\ -0.852 + 0.522i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.4369389594\)
\(L(\frac12)\) \(\approx\) \(0.4369389594\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-9.39e3 + 6.38e3i)T \)
5 \( 1 + (-4.94e5 + 7.20e5i)T \)
good2 \( 1 + (244. - 244. i)T - 1.31e5iT^{2} \)
7 \( 1 + (5.02e6 + 5.02e6i)T + 2.32e14iT^{2} \)
11 \( 1 + 8.18e8iT - 5.05e17T^{2} \)
13 \( 1 + (3.71e9 - 3.71e9i)T - 8.65e18iT^{2} \)
17 \( 1 + (1.47e10 - 1.47e10i)T - 8.27e20iT^{2} \)
19 \( 1 - 1.17e11iT - 5.48e21T^{2} \)
23 \( 1 + (-1.90e10 - 1.90e10i)T + 1.41e23iT^{2} \)
29 \( 1 + 4.44e12T + 7.25e24T^{2} \)
31 \( 1 + 4.29e12T + 2.25e25T^{2} \)
37 \( 1 + (-7.76e12 - 7.76e12i)T + 4.56e26iT^{2} \)
41 \( 1 + 5.45e13iT - 2.61e27T^{2} \)
43 \( 1 + (1.28e13 - 1.28e13i)T - 5.87e27iT^{2} \)
47 \( 1 + (9.98e13 - 9.98e13i)T - 2.66e28iT^{2} \)
53 \( 1 + (-4.28e13 - 4.28e13i)T + 2.05e29iT^{2} \)
59 \( 1 - 1.04e15T + 1.27e30T^{2} \)
61 \( 1 + 2.29e15T + 2.24e30T^{2} \)
67 \( 1 + (-8.72e13 - 8.72e13i)T + 1.10e31iT^{2} \)
71 \( 1 + 7.81e15iT - 2.96e31T^{2} \)
73 \( 1 + (-1.69e15 + 1.69e15i)T - 4.74e31iT^{2} \)
79 \( 1 + 4.69e15iT - 1.81e32T^{2} \)
83 \( 1 + (6.71e15 + 6.71e15i)T + 4.21e32iT^{2} \)
89 \( 1 - 4.06e16T + 1.37e33T^{2} \)
97 \( 1 + (-8.29e15 - 8.29e15i)T + 5.95e33iT^{2} \)
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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

−14.60924896491479912186504262468, −13.32933706206486313348700413949, −12.23129570555457751117805675963, −9.615667923574918082744387481432, −8.757296248406510783066363789843, −7.54880846578693714260177810267, −6.17682469128202540852941466564, −3.72597190675245352973794381743, −1.81051934013846406806242161002, −0.14388037491573258530021360532, 2.14750379069910632445596253633, 2.84734927978431852017857335976, 5.17153402879743477388804212754, 7.32551005587968130213241644502, 9.289295556214107405140378925101, 9.886021023037213281968607598378, 11.04707119619529722674967546234, 13.02557566298047150664256747206, 14.75014814130922238433436407081, 15.23912168055281709760032613145

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