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Properties

Label 2-1512-504.13-c0-0-3
Degree 22
Conductor 15121512
Sign 0.984−0.173i0.984 - 0.173i
Analytic cond. 0.7545860.754586
Root an. cond. 0.8686690.868669
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 0.999·8-s + 0.999·10-s + (1 − 1.73i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 19-s + (0.499 + 0.866i)20-s + (−0.5 + 0.866i)23-s + 1.99·26-s + 0.999·28-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 0.999·8-s + 0.999·10-s + (1 − 1.73i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 19-s + (0.499 + 0.866i)20-s + (−0.5 + 0.866i)23-s + 1.99·26-s + 0.999·28-s + (0.499 − 0.866i)32-s + ⋯

Functional equation

Λ(s)=(1512s/2 ΓC(s) L(s)=((0.984−0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1512s/2 ΓC(s) L(s)=((0.984−0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15121512    =    23⋅33⋅72^{3} \cdot 3^{3} \cdot 7
Sign: 0.984−0.173i0.984 - 0.173i
Analytic conductor: 0.7545860.754586
Root analytic conductor: 0.8686690.868669
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1512(685,⋅)\chi_{1512} (685, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1512, ( :0), 0.984−0.173i)(2,\ 1512,\ (\ :0),\ 0.984 - 0.173i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3894591781.389459178
L(12)L(\frac12) ≈\approx 1.3894591781.389459178
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
3 1 1
7 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good5 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1−T+T2 1 - T + T^{2}
23 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1−T2 1 - T^{2}
59 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1−T+T2 1 - T + T^{2}
73 1−T2 1 - T^{2}
79 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−9.534383599650435449440656322393, −8.776029674428117082901783928761, −7.894102909257364118341306163952, −7.38377621790393367514222971932, −6.22370210577637424572982964102, −5.63176117984537032512651316972, −4.91208139177738474213041289021, −3.77768208479580996045367770927, −3.10484770331262975196641624269, −1.06453178636750602366739496553, 1.72010733574829837955432402525, 2.59959736688575163844704486984, 3.47038124096828908783557826671, 4.44472354494329784016678734980, 5.56229850038318806529436444598, 6.34903460198569765171944728784, 6.76646467344933401893454773905, 8.335262039162833501290609835035, 9.212483071319602567031243023068, 9.638638672057802254118483382762

Graph of the ZZ-function along the critical line