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Properties

Label 2-315-105.104-c1-0-14
Degree 22
Conductor 315315
Sign 0.953+0.302i0.953 + 0.302i
Analytic cond. 2.515282.51528
Root an. cond. 1.585961.58596
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.33·2-s + 3.44·4-s + (1.33 − 1.79i)5-s + (−2.53 − 0.741i)7-s + 3.38·8-s + (3.11 − 4.19i)10-s + 1.41i·11-s + 1.14·13-s + (−5.92 − 1.73i)14-s + 1.00·16-s + 5.20i·17-s + 4.61i·19-s + (4.59 − 6.19i)20-s + 3.30i·22-s − 3.61·23-s + ⋯
L(s)  = 1  + 1.65·2-s + 1.72·4-s + (0.595 − 0.803i)5-s + (−0.959 − 0.280i)7-s + 1.19·8-s + (0.983 − 1.32i)10-s + 0.426i·11-s + 0.316·13-s + (−1.58 − 0.462i)14-s + 0.250·16-s + 1.26i·17-s + 1.05i·19-s + (1.02 − 1.38i)20-s + 0.703i·22-s − 0.754·23-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=((0.953+0.302i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+1/2) L(s)=((0.953+0.302i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: 0.953+0.302i0.953 + 0.302i
Analytic conductor: 2.515282.51528
Root analytic conductor: 1.585961.58596
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ315(314,⋅)\chi_{315} (314, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 315, ( :1/2), 0.953+0.302i)(2,\ 315,\ (\ :1/2),\ 0.953 + 0.302i)

Particular Values

L(1)L(1) ≈\approx 3.01881−0.467227i3.01881 - 0.467227i
L(12)L(\frac12) ≈\approx 3.01881−0.467227i3.01881 - 0.467227i
L(32)L(\frac{3}{2}) 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)
bad3 1 1
5 1+(−1.33+1.79i)T 1 + (-1.33 + 1.79i)T
7 1+(2.53+0.741i)T 1 + (2.53 + 0.741i)T
good2 1−2.33T+2T2 1 - 2.33T + 2T^{2}
11 1−1.41iT−11T2 1 - 1.41iT - 11T^{2}
13 1−1.14T+13T2 1 - 1.14T + 13T^{2}
17 1−5.20iT−17T2 1 - 5.20iT - 17T^{2}
19 1−4.61iT−19T2 1 - 4.61iT - 19T^{2}
23 1+3.61T+23T2 1 + 3.61T + 23T^{2}
29 1−8.34iT−29T2 1 - 8.34iT - 29T^{2}
31 1+8.38iT−31T2 1 + 8.38iT - 31T^{2}
37 1+8.08iT−37T2 1 + 8.08iT - 37T^{2}
41 1−9.19T+41T2 1 - 9.19T + 41T^{2}
43 1+5.11iT−43T2 1 + 5.11iT - 43T^{2}
47 1+1.61iT−47T2 1 + 1.61iT - 47T^{2}
53 1−3.14T+53T2 1 - 3.14T + 53T^{2}
59 1+11.8T+59T2 1 + 11.8T + 59T^{2}
61 1+7.53iT−61T2 1 + 7.53iT - 61T^{2}
67 1−9.57iT−67T2 1 - 9.57iT - 67T^{2}
71 1+5.51iT−71T2 1 + 5.51iT - 71T^{2}
73 1+11.2T+73T2 1 + 11.2T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1+1.61iT−83T2 1 + 1.61iT - 83T^{2}
89 1−7.99T+89T2 1 - 7.99T + 89T^{2}
97 1−1.14T+97T2 1 - 1.14T + 97T^{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

−12.25455869884669842696892408423, −10.85148420112679641784354860048, −9.931811869397251843665305402515, −8.855436667818132341090667412400, −7.43757975257372093541728754913, −6.09783340130460544140036838200, −5.78580806867255059657220861826, −4.37340643847382941214416481915, −3.60647155159479287694005455393, −2.00516954240082135049054479169, 2.59944912669301067306853146054, 3.22832075588624545185506941837, 4.62313823098042159068628163505, 5.85056982641495549265078337060, 6.40039106075702942023232640397, 7.31425679029831873380957148095, 9.072045282538142537233653091890, 10.03562061893904554747608176446, 11.15209335908486735188758483641, 11.83890111523401733795511419794

Graph of the ZZ-function along the critical line