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Properties

Label 2-370-37.10-c1-0-1
Degree $2$
Conductor $370$
Sign $-0.731 - 0.681i$
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 + (1.24 + 2.15i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s − 2.48·6-s + (1.91 + 3.31i)7-s + 0.999·8-s + (−1.59 + 2.76i)9-s − 0.999·10-s − 0.342·11-s + (1.24 − 2.15i)12-s + (−2.26 − 3.93i)13-s − 3.83·14-s + (−1.24 + 2.15i)15-s + (−0.5 + 0.866i)16-s + (3.06 − 5.30i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.718 + 1.24i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s − 1.01·6-s + (0.724 + 1.25i)7-s + 0.353·8-s + (−0.532 + 0.922i)9-s − 0.316·10-s − 0.103·11-s + (0.359 − 0.622i)12-s + (−0.629 − 1.09i)13-s − 1.02·14-s + (−0.321 + 0.556i)15-s + (−0.125 + 0.216i)16-s + (0.742 − 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.731 - 0.681i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.731 - 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.545596 + 1.38605i\)
\(L(\frac12)\) \(\approx\) \(0.545596 + 1.38605i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 5.66i)T \)
good3 \( 1 + (-1.24 - 2.15i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.91 - 3.31i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.342T + 11T^{2} \)
13 \( 1 + (2.26 + 3.93i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.06 + 5.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.74 - 4.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.65T + 23T^{2} \)
29 \( 1 - 0.292T + 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
41 \( 1 + (3.24 + 5.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 9.17T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + (0.474 - 0.822i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.66 + 9.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.85 - 6.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.17 - 8.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.32 - 9.21i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.12T + 73T^{2} \)
79 \( 1 + (0.657 + 1.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.19 + 5.53i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.766 + 1.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.07T + 97T^{2} \)
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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

−11.56491096518743866713403049239, −10.41005533345223686746397034173, −9.771451415927606117033019408173, −9.046898411795897991674933000742, −8.170970948887357632760313553530, −7.33658906443334666181003683302, −5.45000991581499805467339503962, −5.31554251265456536822660981179, −3.62179136425875524702734877258, −2.41405537127740489475931981956, 1.19440734420602831981153952535, 2.08998491759870188754586170695, 3.67370195788944563763667547806, 4.91012851272166238923473581858, 6.69744319898871337664337236241, 7.51393855183252017182887432926, 8.147476450923890333991615343760, 9.116662842569383638701254432458, 10.11362116195166390788866710175, 11.12954360699510270620710260276

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