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Properties

Label 1-5185-5185.19-r0-0-0
Degree $1$
Conductor $5185$
Sign $0.725 - 0.687i$
Analytic cond. $24.0790$
Root an. cond. $24.0790$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.406 + 0.913i)2-s + (0.987 − 0.156i)3-s + (−0.669 + 0.743i)4-s + (0.544 + 0.838i)6-s + (−0.629 + 0.777i)7-s + (−0.951 − 0.309i)8-s + (0.951 − 0.309i)9-s + (−0.707 − 0.707i)11-s + (−0.544 + 0.838i)12-s + (−0.5 + 0.866i)13-s + (−0.965 − 0.258i)14-s + (−0.104 − 0.994i)16-s + (0.669 + 0.743i)18-s + (−0.994 − 0.104i)19-s + (−0.5 + 0.866i)21-s + (0.358 − 0.933i)22-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (0.987 − 0.156i)3-s + (−0.669 + 0.743i)4-s + (0.544 + 0.838i)6-s + (−0.629 + 0.777i)7-s + (−0.951 − 0.309i)8-s + (0.951 − 0.309i)9-s + (−0.707 − 0.707i)11-s + (−0.544 + 0.838i)12-s + (−0.5 + 0.866i)13-s + (−0.965 − 0.258i)14-s + (−0.104 − 0.994i)16-s + (0.669 + 0.743i)18-s + (−0.994 − 0.104i)19-s + (−0.5 + 0.866i)21-s + (0.358 − 0.933i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5185\)    =    \(5 \cdot 17 \cdot 61\)
Sign: $0.725 - 0.687i$
Analytic conductor: \(24.0790\)
Root analytic conductor: \(24.0790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5185} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5185,\ (0:\ ),\ 0.725 - 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6295469461 - 0.2509105545i\)
\(L(\frac12)\) \(\approx\) \(0.6295469461 - 0.2509105545i\)
\(L(1)\) \(\approx\) \(1.019690440 + 0.5744222680i\)
\(L(1)\) \(\approx\) \(1.019690440 + 0.5744222680i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.406 + 0.913i)T \)
3 \( 1 + (0.987 - 0.156i)T \)
7 \( 1 + (-0.629 + 0.777i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.994 - 0.104i)T \)
23 \( 1 + (-0.453 + 0.891i)T \)
29 \( 1 + (-0.965 - 0.258i)T \)
31 \( 1 + (-0.933 + 0.358i)T \)
37 \( 1 + (-0.156 + 0.987i)T \)
41 \( 1 + (0.987 + 0.156i)T \)
43 \( 1 + (0.743 - 0.669i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (0.406 + 0.913i)T \)
67 \( 1 + (-0.978 - 0.207i)T \)
71 \( 1 + (0.544 - 0.838i)T \)
73 \( 1 + (-0.544 + 0.838i)T \)
79 \( 1 + (0.0523 - 0.998i)T \)
83 \( 1 + (-0.406 - 0.913i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.933 - 0.358i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.22490062713795914627993824829, −17.66772332622291337415500222029, −16.67300786129327466655181195082, −15.89146393031355086215102320122, −15.16526940522785418775089873746, −14.559842965692085766972587012596, −14.14339989787219901247470081985, −13.112198325302867579953780181649, −12.76782986693802260797210460229, −12.56036206819204539456576404507, −11.13636023084889472918956694456, −10.543837167770019892803121134096, −10.0917150382140496679764185149, −9.4798979506762103722843525278, −8.80468416662799591817455064108, −7.844596694216100471054738211318, −7.36737556953650832633238856106, −6.36111553560358028308187204489, −5.44513498512669817418630610821, −4.58523271210350365903509609674, −4.00484467474388195476260875518, −3.41105458824518596694435399713, −2.434058470941562284672145763010, −2.20434148359863922768254575284, −0.97598155317231324972879530557, 0.13484548088237242954343181713, 1.83205859172653656856847052401, 2.55855803251658252302641524415, 3.29572916903263846550082649363, 3.94357312582657041334995995770, 4.74603742962050188298394226239, 5.70622448729646509423141054550, 6.17042715018605941744731141092, 7.174516020332796444647528408757, 7.49031129434679285742913321274, 8.575211241884741954278404239529, 8.77912392799180929777073199310, 9.52513021314777541317087649306, 10.206112439551623312377294813, 11.422797859100373140947815260504, 12.19051829049213026374409283274, 12.87038382727032931226202784736, 13.34673531739340986955242981888, 13.939655498431523638792958250716, 14.71302162085338847585765178240, 15.18698797316456507381429939121, 15.78434901126415721812989925681, 16.40141559673791371286081393383, 17.00076439625112766356243696263, 18.08104639298631766771193170027

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