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Properties

Label 2-845-65.18-c1-0-52
Degree 22
Conductor 845845
Sign 0.999−0.0153i0.999 - 0.0153i
Analytic cond. 6.747356.74735
Root an. cond. 2.597562.59756
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  − 0.274i·2-s + (1.67 + 1.67i)3-s + 1.92·4-s + (1.45 − 1.69i)5-s + (0.459 − 0.459i)6-s + 0.386·7-s − 1.07i·8-s + 2.58i·9-s + (−0.466 − 0.399i)10-s + (−3.08 − 3.08i)11-s + (3.21 + 3.21i)12-s − 0.106i·14-s + (5.26 − 0.409i)15-s + 3.55·16-s + (−1.39 − 1.39i)17-s + 0.710·18-s + ⋯
L(s)  = 1  − 0.194i·2-s + (0.964 + 0.964i)3-s + 0.962·4-s + (0.650 − 0.759i)5-s + (0.187 − 0.187i)6-s + 0.145·7-s − 0.381i·8-s + 0.861i·9-s + (−0.147 − 0.126i)10-s + (−0.929 − 0.929i)11-s + (0.928 + 0.928i)12-s − 0.0283i·14-s + (1.36 − 0.105i)15-s + 0.888·16-s + (−0.338 − 0.338i)17-s + 0.167·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((0.999−0.0153i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0153i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((0.999−0.0153i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0153i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: 0.999−0.0153i0.999 - 0.0153i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(408,⋅)\chi_{845} (408, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), 0.999−0.0153i)(2,\ 845,\ (\ :1/2),\ 0.999 - 0.0153i)

Particular Values

L(1)L(1) ≈\approx 2.88989+0.0221645i2.88989 + 0.0221645i
L(12)L(\frac12) ≈\approx 2.88989+0.0221645i2.88989 + 0.0221645i
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)
bad5 1+(−1.45+1.69i)T 1 + (-1.45 + 1.69i)T
13 1 1
good2 1+0.274iT−2T2 1 + 0.274iT - 2T^{2}
3 1+(−1.67−1.67i)T+3iT2 1 + (-1.67 - 1.67i)T + 3iT^{2}
7 1−0.386T+7T2 1 - 0.386T + 7T^{2}
11 1+(3.08+3.08i)T+11iT2 1 + (3.08 + 3.08i)T + 11iT^{2}
17 1+(1.39+1.39i)T+17iT2 1 + (1.39 + 1.39i)T + 17iT^{2}
19 1+(−3.54−3.54i)T+19iT2 1 + (-3.54 - 3.54i)T + 19iT^{2}
23 1+(0.235−0.235i)T−23iT2 1 + (0.235 - 0.235i)T - 23iT^{2}
29 1−8.16iT−29T2 1 - 8.16iT - 29T^{2}
31 1+(2.54−2.54i)T−31iT2 1 + (2.54 - 2.54i)T - 31iT^{2}
37 1−4.82T+37T2 1 - 4.82T + 37T^{2}
41 1+(3.29−3.29i)T−41iT2 1 + (3.29 - 3.29i)T - 41iT^{2}
43 1+(4.82−4.82i)T−43iT2 1 + (4.82 - 4.82i)T - 43iT^{2}
47 1+9.83T+47T2 1 + 9.83T + 47T^{2}
53 1+(7.17+7.17i)T+53iT2 1 + (7.17 + 7.17i)T + 53iT^{2}
59 1+(1.71−1.71i)T−59iT2 1 + (1.71 - 1.71i)T - 59iT^{2}
61 1−10.6T+61T2 1 - 10.6T + 61T^{2}
67 1−6.37iT−67T2 1 - 6.37iT - 67T^{2}
71 1+(3.07−3.07i)T−71iT2 1 + (3.07 - 3.07i)T - 71iT^{2}
73 1−6.08iT−73T2 1 - 6.08iT - 73T^{2}
79 1−3.34iT−79T2 1 - 3.34iT - 79T^{2}
83 1−5.18T+83T2 1 - 5.18T + 83T^{2}
89 1+(3.53−3.53i)T−89iT2 1 + (3.53 - 3.53i)T - 89iT^{2}
97 1+14.7iT−97T2 1 + 14.7iT - 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

−9.994063071058615263260426453591, −9.577333865967520718191297516225, −8.453712234126871896950602116560, −8.053752333057241209463219276928, −6.73028041454928267556159711232, −5.61614223267892496817075050067, −4.85921910199126947393599984166, −3.46226279255571799327655221384, −2.82585707681767409687974846974, −1.52816204998040045852550528471, 1.82649452496407735372914830778, 2.38344948395518168897623577202, 3.23446939484160125320823040752, 5.04117589408957876972175099942, 6.19851799440128426012392063050, 6.90538788343341516028088137002, 7.60757875466253561027510628860, 8.077429191262691468732138574962, 9.372440719181474994592560724719, 10.15121760370021609588218053593

Graph of the ZZ-function along the critical line