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Properties

Label 2-370-185.142-c1-0-6
Degree 22
Conductor 370370
Sign 0.701−0.712i0.701 - 0.712i
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 + (−1 − i)3-s − 4-s + (2 + i)5-s + (1 − i)6-s + (1 + i)7-s − i·8-s − i·9-s + (−1 + 2i)10-s + 2i·11-s + (1 + i)12-s − 2i·13-s + (−1 + i)14-s + (−1 − 3i)15-s + 16-s + 4·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.577 − 0.577i)3-s − 0.5·4-s + (0.894 + 0.447i)5-s + (0.408 − 0.408i)6-s + (0.377 + 0.377i)7-s − 0.353i·8-s − 0.333i·9-s + (−0.316 + 0.632i)10-s + 0.603i·11-s + (0.288 + 0.288i)12-s − 0.554i·13-s + (−0.267 + 0.267i)14-s + (−0.258 − 0.774i)15-s + 0.250·16-s + 0.970·17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.701−0.712i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.701−0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.701−0.712i0.701 - 0.712i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(327,⋅)\chi_{370} (327, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.701−0.712i)(2,\ 370,\ (\ :1/2),\ 0.701 - 0.712i)

Particular Values

L(1)L(1) ≈\approx 1.20964+0.506267i1.20964 + 0.506267i
L(12)L(\frac12) ≈\approx 1.20964+0.506267i1.20964 + 0.506267i
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+(−2−i)T 1 + (-2 - i)T
37 1+(6+i)T 1 + (6 + i)T
good3 1+(1+i)T+3iT2 1 + (1 + i)T + 3iT^{2}
7 1+(−1−i)T+7iT2 1 + (-1 - i)T + 7iT^{2}
11 1−2iT−11T2 1 - 2iT - 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1−4T+17T2 1 - 4T + 17T^{2}
19 1+(−5−5i)T+19iT2 1 + (-5 - 5i)T + 19iT^{2}
23 1−23T2 1 - 23T^{2}
29 1+(−3+3i)T−29iT2 1 + (-3 + 3i)T - 29iT^{2}
31 1+(−7−7i)T+31iT2 1 + (-7 - 7i)T + 31iT^{2}
41 1−41T2 1 - 41T^{2}
43 1+4iT−43T2 1 + 4iT - 43T^{2}
47 1+(7+7i)T+47iT2 1 + (7 + 7i)T + 47iT^{2}
53 1+(−1+i)T−53iT2 1 + (-1 + i)T - 53iT^{2}
59 1+(−1−i)T+59iT2 1 + (-1 - i)T + 59iT^{2}
61 1+(3+3i)T+61iT2 1 + (3 + 3i)T + 61iT^{2}
67 1+(3−3i)T−67iT2 1 + (3 - 3i)T - 67iT^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+(9+9i)T+73iT2 1 + (9 + 9i)T + 73iT^{2}
79 1+(−1−i)T+79iT2 1 + (-1 - i)T + 79iT^{2}
83 1+(5−5i)T−83iT2 1 + (5 - 5i)T - 83iT^{2}
89 1+(5−5i)T−89iT2 1 + (5 - 5i)T - 89iT^{2}
97 1−8T+97T2 1 - 8T + 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

−11.85802598157211148736903372086, −10.31116028768085837893689030303, −9.819277877016164792075999364668, −8.585083425982287523215195122926, −7.51897958150786303110733817438, −6.68526356117287462089483685564, −5.77520787672965429489541320677, −5.15287970719384896425515269860, −3.28734478069087000181489345612, −1.45751200082893960204319637478, 1.22265141455069238326682312772, 2.86806587891364778727409853097, 4.47664800244323080901804971475, 5.14830422622777510526286974851, 6.13613439725025295932208442071, 7.67713971542231896259485145227, 8.823461997305896704784924605521, 9.743408199867838337470935935711, 10.34648853603495862491516881824, 11.28381770311072609603543422432

Graph of the ZZ-function along the critical line