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Properties

Label 2-370-37.36-c1-0-5
Degree $2$
Conductor $370$
Sign $0.775 - 0.630i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 2.24·3-s − 4-s − i·5-s + 2.24i·6-s + 1.52·7-s − i·8-s + 2.05·9-s + 10-s + 2.71·11-s − 2.24·12-s − 0.941i·13-s + 1.52i·14-s − 2.24i·15-s + 16-s + 4.83i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.29·3-s − 0.5·4-s − 0.447i·5-s + 0.918i·6-s + 0.578·7-s − 0.353i·8-s + 0.686·9-s + 0.316·10-s + 0.820·11-s − 0.649·12-s − 0.261i·13-s + 0.408i·14-s − 0.580i·15-s + 0.250·16-s + 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.775 - 0.630i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.775 - 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90194 + 0.675572i\)
\(L(\frac12)\) \(\approx\) \(1.90194 + 0.675572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + iT \)
37 \( 1 + (4.71 - 3.83i)T \)
good3 \( 1 - 2.24T + 3T^{2} \)
7 \( 1 - 1.52T + 7T^{2} \)
11 \( 1 - 2.71T + 11T^{2} \)
13 \( 1 + 0.941iT - 13T^{2} \)
17 \( 1 - 4.83iT - 17T^{2} \)
19 \( 1 - 0.249iT - 19T^{2} \)
23 \( 1 + 0.941iT - 23T^{2} \)
29 \( 1 + 0.719iT - 29T^{2} \)
31 \( 1 + 4.02iT - 31T^{2} \)
41 \( 1 + 8.27T + 41T^{2} \)
43 \( 1 + 2.71iT - 43T^{2} \)
47 \( 1 + 3.30T + 47T^{2} \)
53 \( 1 - 8.39T + 53T^{2} \)
59 \( 1 - 7.30iT - 59T^{2} \)
61 \( 1 + 6.83iT - 61T^{2} \)
67 \( 1 + 7.68T + 67T^{2} \)
71 \( 1 + 3.05T + 71T^{2} \)
73 \( 1 - 9.11T + 73T^{2} \)
79 \( 1 - 1.75iT - 79T^{2} \)
83 \( 1 - 0.131T + 83T^{2} \)
89 \( 1 - 8.99iT - 89T^{2} \)
97 \( 1 + 16.2iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.60601201637352154737814129942, −10.26269405315130133490366140973, −9.305243594999972273512421965601, −8.454535408731731399372285061703, −8.108655394598742955959805405811, −6.93214665642514648815940124060, −5.73125675866419033039564940374, −4.43998740984593056475423863170, −3.46476932458775479750475337363, −1.75434877437549426617228037769, 1.75099906028526191248994240465, 2.92521037072475036209692270690, 3.84498652546583190383149282653, 5.09980658806681565630354670580, 6.77737689717194813490525593769, 7.77576517692004481346599779668, 8.760698868357912906111203207788, 9.339294503359891899860335695549, 10.30541777662529068239911962754, 11.40097255173977897218454835291

Graph of the $Z$-function along the critical line