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Properties

Label 2-370-185.174-c1-0-16
Degree $2$
Conductor $370$
Sign $-0.999 - 0.0176i$
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  + (0.866 − 0.5i)2-s + (−1.73 − i)3-s + (0.499 − 0.866i)4-s + (0.469 + 2.18i)5-s − 1.99·6-s + (−3.73 − 2.15i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (1.5 + 1.65i)10-s − 2.31·11-s + (−1.73 + 0.999i)12-s + (−3.46 − 2i)13-s − 4.31·14-s + (1.37 − 4.25i)15-s + (−0.5 − 0.866i)16-s + (1.14 − 0.658i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.999 − 0.577i)3-s + (0.249 − 0.433i)4-s + (0.210 + 0.977i)5-s − 0.816·6-s + (−1.41 − 0.815i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.474 + 0.524i)10-s − 0.698·11-s + (−0.499 + 0.288i)12-s + (−0.960 − 0.554i)13-s − 1.15·14-s + (0.354 − 1.09i)15-s + (−0.125 − 0.216i)16-s + (0.276 − 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.999 - 0.0176i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.999 - 0.0176i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00448940 + 0.507872i\)
\(L(\frac12)\) \(\approx\) \(0.00448940 + 0.507872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.469 - 2.18i)T \)
37 \( 1 + (-2.59 - 5.5i)T \)
good3 \( 1 + (1.73 + i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.73 + 2.15i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.31T + 11T^{2} \)
13 \( 1 + (3.46 + 2i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.14 + 0.658i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.15 - 2.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 3.31T + 29T^{2} \)
31 \( 1 + 4.31T + 31T^{2} \)
41 \( 1 + (-2.81 + 4.87i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 10.9iT - 43T^{2} \)
47 \( 1 - 2.31iT - 47T^{2} \)
53 \( 1 + (4.01 - 2.31i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.316 + 0.548i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.97 + 6.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.20 - 4.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.15 + 12.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.6iT - 73T^{2} \)
79 \( 1 + (-4.47 + 7.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.74 - 3.31i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.816 + 1.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.94iT - 97T^{2} \)
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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

−10.75635638003759949120132039946, −10.46584079415619489697077047104, −9.592480190666298987855684767074, −7.62307690486296978117592125543, −6.77119080081829537434273628707, −6.23064930373117750662267626020, −5.19313195971648161798496513629, −3.61636346636067079982503763048, −2.58039864961490559864168420799, −0.29182862757285754836733486793, 2.62840298641394066005681522551, 4.20150589882799511009921603555, 5.27071038409650258233254971404, 5.71338397108149616253130791483, 6.75437257163122320545041670051, 8.094902630599746475855522775227, 9.404864320894477942300241454242, 9.828111165715853942733948757186, 11.17230797920728013101051345255, 12.00785796408187313794333454895

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