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Properties

Label 2-15-15.14-c10-0-17
Degree $2$
Conductor $15$
Sign $0.747 + 0.663i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 50.9·2-s + (190. − 150. i)3-s + 1.56e3·4-s + (−553. − 3.07e3i)5-s + (9.72e3 − 7.64e3i)6-s + 8.17e3i·7-s + 2.76e4·8-s + (1.39e4 − 5.73e4i)9-s + (−2.81e4 − 1.56e5i)10-s + 1.97e5i·11-s + (2.99e5 − 2.35e5i)12-s + 2.12e5i·13-s + 4.16e5i·14-s + (−5.67e5 − 5.04e5i)15-s − 1.96e5·16-s + 2.05e6·17-s + ⋯
L(s)  = 1  + 1.59·2-s + (0.785 − 0.618i)3-s + 1.53·4-s + (−0.177 − 0.984i)5-s + (1.25 − 0.983i)6-s + 0.486i·7-s + 0.844·8-s + (0.235 − 0.971i)9-s + (−0.281 − 1.56i)10-s + 1.22i·11-s + (1.20 − 0.946i)12-s + 0.572i·13-s + 0.774i·14-s + (−0.747 − 0.663i)15-s − 0.187·16-s + 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.747 + 0.663i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ 0.747 + 0.663i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(4.42368 - 1.68058i\)
\(L(\frac12)\) \(\approx\) \(4.42368 - 1.68058i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-190. + 150. i)T \)
5 \( 1 + (553. + 3.07e3i)T \)
good2 \( 1 - 50.9T + 1.02e3T^{2} \)
7 \( 1 - 8.17e3iT - 2.82e8T^{2} \)
11 \( 1 - 1.97e5iT - 2.59e10T^{2} \)
13 \( 1 - 2.12e5iT - 1.37e11T^{2} \)
17 \( 1 - 2.05e6T + 2.01e12T^{2} \)
19 \( 1 - 1.39e6T + 6.13e12T^{2} \)
23 \( 1 + 6.13e6T + 4.14e13T^{2} \)
29 \( 1 - 3.53e7iT - 4.20e14T^{2} \)
31 \( 1 + 3.56e7T + 8.19e14T^{2} \)
37 \( 1 + 1.06e8iT - 4.80e15T^{2} \)
41 \( 1 + 1.33e7iT - 1.34e16T^{2} \)
43 \( 1 + 1.18e8iT - 2.16e16T^{2} \)
47 \( 1 - 2.04e7T + 5.25e16T^{2} \)
53 \( 1 - 3.25e8T + 1.74e17T^{2} \)
59 \( 1 + 3.81e8iT - 5.11e17T^{2} \)
61 \( 1 + 4.24e8T + 7.13e17T^{2} \)
67 \( 1 + 1.63e9iT - 1.82e18T^{2} \)
71 \( 1 - 1.20e9iT - 3.25e18T^{2} \)
73 \( 1 + 1.19e9iT - 4.29e18T^{2} \)
79 \( 1 - 3.77e8T + 9.46e18T^{2} \)
83 \( 1 - 1.59e9T + 1.55e19T^{2} \)
89 \( 1 - 8.82e9iT - 3.11e19T^{2} \)
97 \( 1 - 4.83e9iT - 7.37e19T^{2} \)
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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

−16.20402704851205285802739474235, −14.90776612943390568542649661609, −13.95196386878826572617969431286, −12.49046484182774230788321202234, −12.21149175887883206651362748871, −9.214856102331096042059003435242, −7.35092612153897094131462841004, −5.41694829603400572873090901980, −3.77967074275085522171790663365, −1.88632677208891989638000843651, 2.92940248113268945257114474596, 3.83578346160107381666150501566, 5.77849858220690653247259603108, 7.75100247754313723532690052286, 10.20401357692971255679420770819, 11.57833290810052535639793317395, 13.50157107149322817766524963530, 14.21565789418228447761179589281, 15.19495492325516663532978229992, 16.35177478134729645851078666037

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