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Properties

Label 2-370-5.4-c1-0-1
Degree 22
Conductor 370370
Sign −0.635+0.771i-0.635 + 0.771i
Analytic cond. 2.954462.95446
Root an. cond. 1.718851.71885
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  + i·2-s + 2.72i·3-s − 4-s + (−1.42 + 1.72i)5-s − 2.72·6-s − 4.14i·7-s − i·8-s − 4.45·9-s + (−1.72 − 1.42i)10-s − 4.76·11-s − 2.72i·12-s + 3.91i·13-s + 4.14·14-s + (−4.71 − 3.88i)15-s + 16-s + 3.31i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.57i·3-s − 0.5·4-s + (−0.635 + 0.771i)5-s − 1.11·6-s − 1.56i·7-s − 0.353i·8-s − 1.48·9-s + (−0.545 − 0.449i)10-s − 1.43·11-s − 0.788i·12-s + 1.08i·13-s + 1.10·14-s + (−1.21 − 1.00i)15-s + 0.250·16-s + 0.804i·17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((−0.635+0.771i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((−0.635+0.771i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: −0.635+0.771i-0.635 + 0.771i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(149,⋅)\chi_{370} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), −0.635+0.771i)(2,\ 370,\ (\ :1/2),\ -0.635 + 0.771i)

Particular Values

L(1)L(1) ≈\approx 0.278007−0.589096i0.278007 - 0.589096i
L(12)L(\frac12) ≈\approx 0.278007−0.589096i0.278007 - 0.589096i
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)
bad2 1−iT 1 - iT
5 1+(1.42−1.72i)T 1 + (1.42 - 1.72i)T
37 1−iT 1 - iT
good3 1−2.72iT−3T2 1 - 2.72iT - 3T^{2}
7 1+4.14iT−7T2 1 + 4.14iT - 7T^{2}
11 1+4.76T+11T2 1 + 4.76T + 11T^{2}
13 1−3.91iT−13T2 1 - 3.91iT - 13T^{2}
17 1−3.31iT−17T2 1 - 3.31iT - 17T^{2}
19 1−1.85T+19T2 1 - 1.85T + 19T^{2}
23 1−1.54iT−23T2 1 - 1.54iT - 23T^{2}
29 1+8.87T+29T2 1 + 8.87T + 29T^{2}
31 1−9.75T+31T2 1 - 9.75T + 31T^{2}
41 1+5.06T+41T2 1 + 5.06T + 41T^{2}
43 1−9.99iT−43T2 1 - 9.99iT - 43T^{2}
47 1−4.82iT−47T2 1 - 4.82iT - 47T^{2}
53 1−5.13iT−53T2 1 - 5.13iT - 53T^{2}
59 1−1.05T+59T2 1 - 1.05T + 59T^{2}
61 1+4.14T+61T2 1 + 4.14T + 61T^{2}
67 1−1.00iT−67T2 1 - 1.00iT - 67T^{2}
71 1−6.45T+71T2 1 - 6.45T + 71T^{2}
73 1+10.7iT−73T2 1 + 10.7iT - 73T^{2}
79 1−1.19T+79T2 1 - 1.19T + 79T^{2}
83 1+10.6iT−83T2 1 + 10.6iT - 83T^{2}
89 1+7.29T+89T2 1 + 7.29T + 89T^{2}
97 1−14.8iT−97T2 1 - 14.8iT - 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

−11.55600511943335852781434535284, −10.71903331077717451307441586625, −10.25989179603639685955839769232, −9.426149915438756562736027514646, −8.068916918924450628884720247867, −7.42567522658600133985910479644, −6.27044265482744543734007670000, −4.84058939518166643161312483564, −4.15397684502326706821031253525, −3.28305222794189869118810119477, 0.42785923145299655776089429301, 2.12798534705374300661897601621, 3.03447413166692801916792642754, 5.15036458068803691678015627947, 5.67229176060720202403518309978, 7.29277915265731209765715638638, 8.178917193068714702186877004839, 8.616853468119460071405087650975, 9.877219962124118192842413110611, 11.27385290597049912506484471065

Graph of the ZZ-function along the critical line