↑

Properties

Label 2-845-65.47-c1-0-21
Degree 22
Conductor 845845
Sign 0.383−0.923i0.383 - 0.923i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.131i·2-s + (−0.243 + 0.243i)3-s + 1.98·4-s + (−2.08 − 0.813i)5-s + (−0.0319 − 0.0319i)6-s + 2.78·7-s + 0.522i·8-s + 2.88i·9-s + (0.106 − 0.273i)10-s + (−2.86 + 2.86i)11-s + (−0.482 + 0.482i)12-s + 0.365i·14-s + (0.704 − 0.308i)15-s + 3.89·16-s + (−1.71 + 1.71i)17-s − 0.378·18-s + ⋯
L(s)  = 1  + 0.0928i·2-s + (−0.140 + 0.140i)3-s + 0.991·4-s + (−0.931 − 0.363i)5-s + (−0.0130 − 0.0130i)6-s + 1.05·7-s + 0.184i·8-s + 0.960i·9-s + (0.0337 − 0.0864i)10-s + (−0.864 + 0.864i)11-s + (−0.139 + 0.139i)12-s + 0.0976i·14-s + (0.181 − 0.0797i)15-s + 0.974·16-s + (−0.415 + 0.415i)17-s − 0.0891·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((0.383−0.923i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((0.383−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: 0.383−0.923i0.383 - 0.923i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(437,⋅)\chi_{845} (437, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), 0.383−0.923i)(2,\ 845,\ (\ :1/2),\ 0.383 - 0.923i)

Particular Values

L(1)L(1) ≈\approx 1.34481+0.897649i1.34481 + 0.897649i
L(12)L(\frac12) ≈\approx 1.34481+0.897649i1.34481 + 0.897649i
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+(2.08+0.813i)T 1 + (2.08 + 0.813i)T
13 1 1
good2 1−0.131iT−2T2 1 - 0.131iT - 2T^{2}
3 1+(0.243−0.243i)T−3iT2 1 + (0.243 - 0.243i)T - 3iT^{2}
7 1−2.78T+7T2 1 - 2.78T + 7T^{2}
11 1+(2.86−2.86i)T−11iT2 1 + (2.86 - 2.86i)T - 11iT^{2}
17 1+(1.71−1.71i)T−17iT2 1 + (1.71 - 1.71i)T - 17iT^{2}
19 1+(1.34−1.34i)T−19iT2 1 + (1.34 - 1.34i)T - 19iT^{2}
23 1+(−5.64−5.64i)T+23iT2 1 + (-5.64 - 5.64i)T + 23iT^{2}
29 1+4.57iT−29T2 1 + 4.57iT - 29T^{2}
31 1+(3.87+3.87i)T+31iT2 1 + (3.87 + 3.87i)T + 31iT^{2}
37 1−7.01T+37T2 1 - 7.01T + 37T^{2}
41 1+(−4.54−4.54i)T+41iT2 1 + (-4.54 - 4.54i)T + 41iT^{2}
43 1+(−4.57−4.57i)T+43iT2 1 + (-4.57 - 4.57i)T + 43iT^{2}
47 1−0.512T+47T2 1 - 0.512T + 47T^{2}
53 1+(1.32−1.32i)T−53iT2 1 + (1.32 - 1.32i)T - 53iT^{2}
59 1+(−1.85−1.85i)T+59iT2 1 + (-1.85 - 1.85i)T + 59iT^{2}
61 1+1.28T+61T2 1 + 1.28T + 61T^{2}
67 1+3.61iT−67T2 1 + 3.61iT - 67T^{2}
71 1+(4.54+4.54i)T+71iT2 1 + (4.54 + 4.54i)T + 71iT^{2}
73 1−9.93iT−73T2 1 - 9.93iT - 73T^{2}
79 1+8.37iT−79T2 1 + 8.37iT - 79T^{2}
83 1+3.17T+83T2 1 + 3.17T + 83T^{2}
89 1+(4.40+4.40i)T+89iT2 1 + (4.40 + 4.40i)T + 89iT^{2}
97 1−11.7iT−97T2 1 - 11.7iT - 97T^{2}
show more
show less
   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

−10.71879285173691428314109314364, −9.591797208353423134906707584190, −8.273448355347056787999059571557, −7.70328811411742297133289807586, −7.32449758366342254257227136998, −5.87312361146268882647081836295, −4.96339269911374488579681077609, −4.22850449951973310771153010657, −2.67879317249199878433806318932, −1.61692529841535266667236747447, 0.824574418863889243187708578200, 2.51483757303268521519839197403, 3.38474056406518403078066307100, 4.62299766618506117821146338125, 5.73469106416047515754200568845, 6.80127565349326840929840607186, 7.33218642810825614187829507748, 8.271637686287766976609188251552, 8.967666307238778093806198048885, 10.49546598175237808356316417431

Graph of the ZZ-function along the critical line