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Properties

Label 2-370-185.184-c1-0-16
Degree $2$
Conductor $370$
Sign $0.444 + 0.895i$
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  + 2-s − 1.76i·3-s + 4-s + (1.62 − 1.53i)5-s − 1.76i·6-s + 1.22i·7-s + 8-s − 0.105·9-s + (1.62 − 1.53i)10-s + 1.87·11-s − 1.76i·12-s − 6.50·13-s + 1.22i·14-s + (−2.69 − 2.86i)15-s + 16-s + 0.765·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.01i·3-s + 0.5·4-s + (0.728 − 0.685i)5-s − 0.719i·6-s + 0.461i·7-s + 0.353·8-s − 0.0350·9-s + (0.515 − 0.484i)10-s + 0.564·11-s − 0.508i·12-s − 1.80·13-s + 0.326i·14-s + (−0.697 − 0.741i)15-s + 0.250·16-s + 0.185·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.444 + 0.895i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90443 - 1.18142i\)
\(L(\frac12)\) \(\approx\) \(1.90443 - 1.18142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (-1.62 + 1.53i)T \)
37 \( 1 + (1.76 - 5.82i)T \)
good3 \( 1 + 1.76iT - 3T^{2} \)
7 \( 1 - 1.22iT - 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 + 6.50T + 13T^{2} \)
17 \( 1 - 0.765T + 17T^{2} \)
19 \( 1 - 3.34iT - 19T^{2} \)
23 \( 1 + 1.38T + 23T^{2} \)
29 \( 1 + 1.72iT - 29T^{2} \)
31 \( 1 - 4.11iT - 31T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 + 4.91T + 43T^{2} \)
47 \( 1 - 6.30iT - 47T^{2} \)
53 \( 1 - 2.57iT - 53T^{2} \)
59 \( 1 - 10.5iT - 59T^{2} \)
61 \( 1 + 11.1iT - 61T^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 - 0.963T + 71T^{2} \)
73 \( 1 + 9.03iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 - 0.00656iT - 83T^{2} \)
89 \( 1 + 4.70iT - 89T^{2} \)
97 \( 1 - 0.403T + 97T^{2} \)
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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

−11.86093340750452193888347191881, −10.23389193196646498259840207705, −9.489498275335560789124178146819, −8.250655663535713398744053086094, −7.27694485012888885329515779372, −6.36338720790051584285765276987, −5.43526383536633173332637457084, −4.41942860194813773220038444717, −2.59021434428725727333957746013, −1.52803476626265972983399746203, 2.28383345846500085769172630406, 3.56299882004999211712185027639, 4.60963799439051250423538946473, 5.47191555736213896407473971784, 6.77224566899951745240078227598, 7.44089631565782522281838412649, 9.221278506919131908520650029578, 9.908061832052018599021969020426, 10.54283186446143274280954543083, 11.44977352175539014063502385943

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