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Properties

Label 2-15-15.2-c17-0-31
Degree $2$
Conductor $15$
Sign $0.485 - 0.874i$
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  + (398. − 398. i)2-s + (−6.49e3 − 9.32e3i)3-s − 1.86e5i·4-s + (−8.69e5 − 8.57e4i)5-s + (−6.29e6 − 1.12e6i)6-s + (−6.71e6 − 6.71e6i)7-s + (−2.18e7 − 2.18e7i)8-s + (−4.47e7 + 1.21e8i)9-s + (−3.80e8 + 3.11e8i)10-s + 5.34e7i·11-s + (−1.73e9 + 1.20e9i)12-s + (2.34e9 − 2.34e9i)13-s − 5.34e9·14-s + (4.84e9 + 8.66e9i)15-s + 6.95e9·16-s + (−3.10e10 + 3.10e10i)17-s + ⋯
L(s)  = 1  + (1.09 − 1.09i)2-s + (−0.571 − 0.820i)3-s − 1.41i·4-s + (−0.995 − 0.0981i)5-s + (−1.53 − 0.273i)6-s + (−0.440 − 0.440i)7-s + (−0.461 − 0.461i)8-s + (−0.346 + 0.938i)9-s + (−1.20 + 0.986i)10-s + 0.0751i·11-s + (−1.16 + 0.811i)12-s + (0.796 − 0.796i)13-s − 0.967·14-s + (0.488 + 0.872i)15-s + 0.404·16-s + (−1.07 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.485 - 0.874i$
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: \((2,\ 15,\ (\ :17/2),\ 0.485 - 0.874i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.3256145706\)
\(L(\frac12)\) \(\approx\) \(0.3256145706\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.49e3 + 9.32e3i)T \)
5 \( 1 + (8.69e5 + 8.57e4i)T \)
good2 \( 1 + (-398. + 398. i)T - 1.31e5iT^{2} \)
7 \( 1 + (6.71e6 + 6.71e6i)T + 2.32e14iT^{2} \)
11 \( 1 - 5.34e7iT - 5.05e17T^{2} \)
13 \( 1 + (-2.34e9 + 2.34e9i)T - 8.65e18iT^{2} \)
17 \( 1 + (3.10e10 - 3.10e10i)T - 8.27e20iT^{2} \)
19 \( 1 + 1.76e10iT - 5.48e21T^{2} \)
23 \( 1 + (1.25e11 + 1.25e11i)T + 1.41e23iT^{2} \)
29 \( 1 + 3.24e12T + 7.25e24T^{2} \)
31 \( 1 + 1.46e12T + 2.25e25T^{2} \)
37 \( 1 + (-8.50e12 - 8.50e12i)T + 4.56e26iT^{2} \)
41 \( 1 - 9.41e13iT - 2.61e27T^{2} \)
43 \( 1 + (8.91e13 - 8.91e13i)T - 5.87e27iT^{2} \)
47 \( 1 + (-4.64e13 + 4.64e13i)T - 2.66e28iT^{2} \)
53 \( 1 + (2.24e14 + 2.24e14i)T + 2.05e29iT^{2} \)
59 \( 1 + 2.10e15T + 1.27e30T^{2} \)
61 \( 1 + 2.19e15T + 2.24e30T^{2} \)
67 \( 1 + (-1.54e15 - 1.54e15i)T + 1.10e31iT^{2} \)
71 \( 1 + 9.50e15iT - 2.96e31T^{2} \)
73 \( 1 + (4.29e15 - 4.29e15i)T - 4.74e31iT^{2} \)
79 \( 1 + 6.78e15iT - 1.81e32T^{2} \)
83 \( 1 + (4.70e15 + 4.70e15i)T + 4.21e32iT^{2} \)
89 \( 1 + 2.20e16T + 1.37e33T^{2} \)
97 \( 1 + (-2.95e16 - 2.95e16i)T + 5.95e33iT^{2} \)
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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

−13.30503240934995938212988665162, −12.77849837976522478007614739611, −11.47263749517908189390370146290, −10.68609323571824846663627844902, −8.003495887354181708773250296160, −6.24920575435639216400190370327, −4.56028241017550818427696501490, −3.23813586126465340532206609398, −1.50271389419571206347037135629, −0.079403341014375122326511358702, 3.51420935623482184802613366523, 4.49361510481315233634853961392, 5.87245469129191205947434320902, 7.11238055911920764832951743263, 9.012805606943749613345663896726, 11.13260241569035594958601992295, 12.31910099154916695847207814294, 13.93351019092053182999826244406, 15.37556454754978167040649491024, 15.82816027168711829011526864936

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