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Properties

Label 2-370-37.26-c1-0-0
Degree $2$
Conductor $370$
Sign $-0.993 - 0.115i$
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 + (−1.38 + 2.40i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 2.77·6-s + (−1.65 + 2.86i)7-s + 0.999·8-s + (−2.35 − 4.08i)9-s − 0.999·10-s + 1.52·11-s + (−1.38 − 2.40i)12-s + (−2.09 + 3.62i)13-s + 3.30·14-s + (1.38 + 2.40i)15-s + (−0.5 − 0.866i)16-s + (−3.90 − 6.76i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.802 + 1.38i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + 1.13·6-s + (−0.625 + 1.08i)7-s + 0.353·8-s + (−0.786 − 1.36i)9-s − 0.316·10-s + 0.461·11-s + (−0.401 − 0.694i)12-s + (−0.581 + 1.00i)13-s + 0.884·14-s + (0.358 + 0.621i)15-s + (−0.125 − 0.216i)16-s + (−0.946 − 1.63i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.993 - 0.115i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.993 - 0.115i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0164852 + 0.285005i\)
\(L(\frac12)\) \(\approx\) \(0.0164852 + 0.285005i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (2.87 - 5.36i)T \)
good3 \( 1 + (1.38 - 2.40i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.65 - 2.86i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.52T + 11T^{2} \)
13 \( 1 + (2.09 - 3.62i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.90 + 6.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.110 + 0.191i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.52T + 23T^{2} \)
29 \( 1 + 6.49T + 29T^{2} \)
31 \( 1 + 4.41T + 31T^{2} \)
41 \( 1 + (0.610 - 1.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 0.162T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + (-1.98 - 3.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.543 + 0.940i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.24 + 12.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.83 - 6.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.08 - 12.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.80T + 73T^{2} \)
79 \( 1 + (2.52 - 4.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.71 - 8.17i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.66 - 15.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.38T + 97T^{2} \)
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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

−11.73670034856786985463734047407, −10.98102386334011149549404219809, −9.774787669171032875843699707645, −9.373769534920697892430461154065, −8.837859671579324453453144079536, −6.98766700914879932548003221969, −5.75348154401639637230193194635, −4.86296514984672649015973965207, −3.89962660111783910739408884010, −2.40628554290662796725594291143, 0.22773485841677310350487891455, 1.84914645517564051397878052737, 3.91203350871080451544902344016, 5.68066756347350282256813898930, 6.27278864180423343487412863745, 7.21557183909617592970986467237, 7.61853496126903900194905281901, 8.933377597502440625277304686583, 10.33268056017700519764503683166, 10.72646302941709621525014874073

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