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Properties

Label 2-185-37.26-c1-0-9
Degree $2$
Conductor $185$
Sign $0.839 - 0.543i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
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  + (0.704 + 1.22i)2-s + (0.0771 − 0.133i)3-s + (0.00725 − 0.0125i)4-s + (0.5 − 0.866i)5-s + 0.217·6-s + (1.20 − 2.08i)7-s + 2.83·8-s + (1.48 + 2.57i)9-s + 1.40·10-s − 4.38·11-s + (−0.00112 − 0.00194i)12-s + (−1.99 + 3.45i)13-s + 3.39·14-s + (−0.0771 − 0.133i)15-s + (1.98 + 3.43i)16-s + (−0.776 − 1.34i)17-s + ⋯
L(s)  = 1  + (0.498 + 0.862i)2-s + (0.0445 − 0.0771i)3-s + (0.00362 − 0.00628i)4-s + (0.223 − 0.387i)5-s + 0.0887·6-s + (0.455 − 0.789i)7-s + 1.00·8-s + (0.496 + 0.859i)9-s + 0.445·10-s − 1.32·11-s + (−0.000323 − 0.000560i)12-s + (−0.553 + 0.958i)13-s + 0.908·14-s + (−0.0199 − 0.0345i)15-s + (0.496 + 0.859i)16-s + (−0.188 − 0.326i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59487 + 0.471185i\)
\(L(\frac12)\) \(\approx\) \(1.59487 + 0.471185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (5.76 + 1.94i)T \)
good2 \( 1 + (-0.704 - 1.22i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.0771 + 0.133i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.20 + 2.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.38T + 11T^{2} \)
13 \( 1 + (1.99 - 3.45i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.776 + 1.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.736 + 1.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 9.41T + 23T^{2} \)
29 \( 1 - 3.09T + 29T^{2} \)
31 \( 1 - 8.30T + 31T^{2} \)
41 \( 1 + (4.05 - 7.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 2.66T + 43T^{2} \)
47 \( 1 - 1.40T + 47T^{2} \)
53 \( 1 + (4.36 + 7.55i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.94 + 6.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.760 - 1.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.41 + 4.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.93 - 8.54i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + (-8.24 + 14.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.26 - 5.65i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.38 - 11.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−13.17870625628928964155406457158, −11.79994353725463401642313987807, −10.52346526766120987100849100845, −9.973471371716149004850566890360, −8.140954976677764430170330925893, −7.53209561840585063492817972263, −6.44383800008400463420886822294, −5.02160472776939487970167299630, −4.51356098745868371578956120085, −1.98075743165114652958384504085, 2.17667146703163883836235564393, 3.26872563980173923392221758344, 4.71945214014627878515087592482, 5.95127477323873022793471822748, 7.49798004102765429300932193158, 8.405693459437542684182970008569, 10.14866127610512453507623055755, 10.38741750375143205452774079873, 11.92851092230026363119421515507, 12.23914069821522524893017576949

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