Properties

Label 2-18515-1.1-c1-0-5
Degree $2$
Conductor $18515$
Sign $1$
Analytic cond. $147.843$
Root an. cond. $12.1590$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s − 4-s − 5-s − 3·6-s − 7-s + 3·8-s + 6·9-s + 10-s − 11-s − 3·12-s − 2·13-s + 14-s − 3·15-s − 16-s + 3·17-s − 6·18-s + 4·19-s + 20-s − 3·21-s + 22-s + 9·24-s + 25-s + 2·26-s + 9·27-s + 28-s + 10·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s − 1/2·4-s − 0.447·5-s − 1.22·6-s − 0.377·7-s + 1.06·8-s + 2·9-s + 0.316·10-s − 0.301·11-s − 0.866·12-s − 0.554·13-s + 0.267·14-s − 0.774·15-s − 1/4·16-s + 0.727·17-s − 1.41·18-s + 0.917·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s + 1.83·24-s + 1/5·25-s + 0.392·26-s + 1.73·27-s + 0.188·28-s + 1.85·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18515 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18515 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18515\)    =    \(5 \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(147.843\)
Root analytic conductor: \(12.1590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{18515} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 18515,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.373418184\)
\(L(\frac12)\) \(\approx\) \(2.373418184\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 - p T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−15.83751759716204, −15.22253892106027, −14.45610136948231, −14.20300308187880, −13.75063673276947, −13.12944991598451, −12.55813372352819, −12.13153185516502, −11.15645605126262, −10.32673097944919, −10.00977032590697, −9.445857156424017, −8.986014396559306, −8.471686393517889, −7.849440734397700, −7.521533808395293, −7.064458975596105, −5.966060466262354, −5.014610053759698, −4.377581598993532, −3.797397619023154, −2.992862742168567, −2.595668857876958, −1.481845998718128, −0.7047538416924796, 0.7047538416924796, 1.481845998718128, 2.595668857876958, 2.992862742168567, 3.797397619023154, 4.377581598993532, 5.014610053759698, 5.966060466262354, 7.064458975596105, 7.521533808395293, 7.849440734397700, 8.471686393517889, 8.986014396559306, 9.445857156424017, 10.00977032590697, 10.32673097944919, 11.15645605126262, 12.13153185516502, 12.55813372352819, 13.12944991598451, 13.75063673276947, 14.20300308187880, 14.45610136948231, 15.22253892106027, 15.83751759716204

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