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Properties

Label 2-370-185.103-c1-0-9
Degree $2$
Conductor $370$
Sign $0.482 + 0.875i$
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  + (−0.5 − 0.866i)2-s + (0.133 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (−1.86 + 1.23i)5-s + (0.366 − 0.366i)6-s + (−0.732 − 2.73i)7-s + 0.999·8-s + (2.36 − 1.36i)9-s + (2 + i)10-s + 2i·11-s + (−0.499 − 0.133i)12-s + (2.23 − 3.86i)13-s + (−1.99 + 2i)14-s + (−0.866 − 0.767i)15-s + (−0.5 − 0.866i)16-s + (3.63 − 2.09i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.0773 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.834 + 0.550i)5-s + (0.149 − 0.149i)6-s + (−0.276 − 1.03i)7-s + 0.353·8-s + (0.788 − 0.455i)9-s + (0.632 + 0.316i)10-s + 0.603i·11-s + (−0.144 − 0.0386i)12-s + (0.619 − 1.07i)13-s + (−0.534 + 0.534i)14-s + (−0.223 − 0.198i)15-s + (−0.125 − 0.216i)16-s + (0.881 − 0.508i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.482 + 0.875i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.482 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881314 - 0.520761i\)
\(L(\frac12)\) \(\approx\) \(0.881314 - 0.520761i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (1.86 - 1.23i)T \)
37 \( 1 + (5.5 - 2.59i)T \)
good3 \( 1 + (-0.133 - 0.5i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.732 + 2.73i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (-2.23 + 3.86i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.63 + 2.09i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.36 + 0.901i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (1.26 - 1.26i)T - 29iT^{2} \)
31 \( 1 + (1.83 + 1.83i)T + 31iT^{2} \)
41 \( 1 + (0.401 + 0.232i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 11T + 43T^{2} \)
47 \( 1 + (-4.19 + 4.19i)T - 47iT^{2} \)
53 \( 1 + (-3.13 + 11.6i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.09 - 7.83i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.46 - 1.73i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-13.5 + 3.63i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.46 - 9.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + (6.83 - 1.83i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (3.43 - 12.8i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-12.5 - 3.36i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 5.46iT - 97T^{2} \)
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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

−11.08296130940835369449460601239, −10.26212643897124940412880352351, −9.797582429897144429768776972096, −8.506669435061176224482599103258, −7.36364614950166980323497024033, −6.97707075817635144948493727973, −5.03973352116248190176634666391, −3.76554601353263863144179051282, −3.21479131348686224121636395301, −0.935111060567454286714856075917, 1.44099208712006002643621054645, 3.46771891406794368239209058427, 4.82032778253639872022704033537, 5.82956848961958141155756559045, 6.97669237460714715250315086268, 7.84669274081683740491010001061, 8.739780769352797807812822476756, 9.319435132109838122293877029716, 10.61659865487924380626231554860, 11.65251061247566724089060680989

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