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Properties

Label 2-15-15.2-c5-0-1
Degree $2$
Conductor $15$
Sign $-0.910 + 0.413i$
Analytic cond. $2.40575$
Root an. cond. $1.55105$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.77 + 6.77i)2-s + (3.70 + 15.1i)3-s − 59.9i·4-s + (−52.1 − 20.0i)5-s + (−127. − 77.5i)6-s + (47.9 + 47.9i)7-s + (189. + 189. i)8-s + (−215. + 112. i)9-s + (489. − 217. i)10-s + 234. i·11-s + (907. − 221. i)12-s + (−723. + 723. i)13-s − 650.·14-s + (110. − 864. i)15-s − 649.·16-s + (335. − 335. i)17-s + ⋯
L(s)  = 1  + (−1.19 + 1.19i)2-s + (0.237 + 0.971i)3-s − 1.87i·4-s + (−0.933 − 0.359i)5-s + (−1.44 − 0.879i)6-s + (0.370 + 0.370i)7-s + (1.04 + 1.04i)8-s + (−0.887 + 0.461i)9-s + (1.54 − 0.687i)10-s + 0.585i·11-s + (1.81 − 0.444i)12-s + (−1.18 + 1.18i)13-s − 0.886·14-s + (0.127 − 0.991i)15-s − 0.633·16-s + (0.281 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.910 + 0.413i$
Analytic conductor: \(2.40575\)
Root analytic conductor: \(1.55105\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5/2),\ -0.910 + 0.413i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.100899 - 0.466458i\)
\(L(\frac12)\) \(\approx\) \(0.100899 - 0.466458i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.70 - 15.1i)T \)
5 \( 1 + (52.1 + 20.0i)T \)
good2 \( 1 + (6.77 - 6.77i)T - 32iT^{2} \)
7 \( 1 + (-47.9 - 47.9i)T + 1.68e4iT^{2} \)
11 \( 1 - 234. iT - 1.61e5T^{2} \)
13 \( 1 + (723. - 723. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-335. + 335. i)T - 1.41e6iT^{2} \)
19 \( 1 + 92.0iT - 2.47e6T^{2} \)
23 \( 1 + (-2.14e3 - 2.14e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 996.T + 2.05e7T^{2} \)
31 \( 1 - 2.52e3T + 2.86e7T^{2} \)
37 \( 1 + (-2.46e3 - 2.46e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 2.75e3iT - 1.15e8T^{2} \)
43 \( 1 + (6.54e3 - 6.54e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (449. - 449. i)T - 2.29e8iT^{2} \)
53 \( 1 + (-9.02e3 - 9.02e3i)T + 4.18e8iT^{2} \)
59 \( 1 - 5.80e3T + 7.14e8T^{2} \)
61 \( 1 - 2.71e4T + 8.44e8T^{2} \)
67 \( 1 + (4.35e4 + 4.35e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 5.54e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.61e4 - 2.61e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 2.37e4iT - 3.07e9T^{2} \)
83 \( 1 + (6.35e4 + 6.35e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 2.37e4T + 5.58e9T^{2} \)
97 \( 1 + (-8.07e4 - 8.07e4i)T + 8.58e9iT^{2} \)
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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

−18.96372580226461342964861499114, −17.30744678450370299460702215918, −16.42020459733971603083184815413, −15.36253074782372232996009573645, −14.58900775038885582937816621412, −11.65760171122092781970745295187, −9.790910915028547281459818830835, −8.726700322860298188403224234389, −7.35740858159864348173369900188, −4.89784831786619883757277028005, 0.57828508363726777604921313751, 2.93164148835353575677945934522, 7.48771151644215281961914583141, 8.462489008115812858846427003873, 10.48019260237190394658731631649, 11.69541955912700556894938078041, 12.75134734563734052226010551911, 14.72638109904039325616815858995, 16.94286551808512777075163290465, 18.01582601511313499395423806716

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