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Properties

Label 2-370-185.103-c1-0-18
Degree $2$
Conductor $370$
Sign $-0.779 + 0.626i$
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 + (−0.875 − 3.26i)3-s + (−0.499 + 0.866i)4-s + (−0.0954 − 2.23i)5-s + (2.39 − 2.39i)6-s + (−0.371 − 1.38i)7-s − 0.999·8-s + (−7.30 + 4.21i)9-s + (1.88 − 1.19i)10-s + 3.50i·11-s + (3.26 + 0.875i)12-s + (2.36 − 4.09i)13-s + (1.01 − 1.01i)14-s + (−7.21 + 2.26i)15-s + (−0.5 − 0.866i)16-s + (−0.841 + 0.485i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.505 − 1.88i)3-s + (−0.249 + 0.433i)4-s + (−0.0426 − 0.999i)5-s + (0.976 − 0.976i)6-s + (−0.140 − 0.524i)7-s − 0.353·8-s + (−2.43 + 1.40i)9-s + (0.596 − 0.379i)10-s + 1.05i·11-s + (0.942 + 0.252i)12-s + (0.655 − 1.13i)13-s + (0.271 − 0.271i)14-s + (−1.86 + 0.585i)15-s + (−0.125 − 0.216i)16-s + (−0.204 + 0.117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.779 + 0.626i$
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: \((2,\ 370,\ (\ :1/2),\ -0.779 + 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.321499 - 0.913686i\)
\(L(\frac12)\) \(\approx\) \(0.321499 - 0.913686i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0954 + 2.23i)T \)
37 \( 1 + (-2.45 + 5.56i)T \)
good3 \( 1 + (0.875 + 3.26i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.371 + 1.38i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 3.50iT - 11T^{2} \)
13 \( 1 + (-2.36 + 4.09i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.841 - 0.485i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.83 - 0.760i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 4.40T + 23T^{2} \)
29 \( 1 + (-7.48 + 7.48i)T - 29iT^{2} \)
31 \( 1 + (3.66 + 3.66i)T + 31iT^{2} \)
41 \( 1 + (-3.39 - 1.96i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 2.05T + 43T^{2} \)
47 \( 1 + (-7.19 + 7.19i)T - 47iT^{2} \)
53 \( 1 + (0.343 - 1.28i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.29 + 4.84i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.28 + 0.344i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.91 + 0.513i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.83 - 6.64i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.69 - 8.69i)T - 73iT^{2} \)
79 \( 1 + (2.35 - 0.631i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-0.907 + 3.38i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-8.16 - 2.18i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 4.02iT - 97T^{2} \)
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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

−11.49705094538963735105122505833, −10.18819499074335260026084413841, −8.619120083360392615010260408546, −7.950730445658604248787557966480, −7.27523345491428964060129313836, −6.19999186120819848762848146546, −5.55961413386583742431452995255, −4.23041874355460554778323209947, −2.17651014889182905295210885593, −0.61346945575133739303404582202, 2.82217666942902605045886917310, 3.68776487541639949258360188414, 4.59609825882042471545699617689, 5.83807156823019570235599425752, 6.43260287448611545761315955255, 8.655552998652650796438658344256, 9.184868255381067794935276960174, 10.32460446991587429484770500641, 10.77762443415458436179146681002, 11.45786033986505612352837122417

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