Properties

Label 2-230-1.1-c3-0-21
Degree $2$
Conductor $230$
Sign $-1$
Analytic cond. $13.5704$
Root an. cond. $3.68380$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 4·4-s − 5·5-s + 2·6-s − 18·7-s + 8·8-s − 26·9-s − 10·10-s − 32·11-s + 4·12-s − 47·13-s − 36·14-s − 5·15-s + 16·16-s + 20·17-s − 52·18-s + 36·19-s − 20·20-s − 18·21-s − 64·22-s − 23·23-s + 8·24-s + 25·25-s − 94·26-s − 53·27-s − 72·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.192·3-s + 1/2·4-s − 0.447·5-s + 0.136·6-s − 0.971·7-s + 0.353·8-s − 0.962·9-s − 0.316·10-s − 0.877·11-s + 0.0962·12-s − 1.00·13-s − 0.687·14-s − 0.0860·15-s + 1/4·16-s + 0.285·17-s − 0.680·18-s + 0.434·19-s − 0.223·20-s − 0.187·21-s − 0.620·22-s − 0.208·23-s + 0.0680·24-s + 1/5·25-s − 0.709·26-s − 0.377·27-s − 0.485·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{230} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
5 \( 1 + p T \)
23 \( 1 + p T \)
good3 \( 1 - T + p^{3} T^{2} \)
7 \( 1 + 18 T + p^{3} T^{2} \)
11 \( 1 + 32 T + p^{3} T^{2} \)
13 \( 1 + 47 T + p^{3} T^{2} \)
17 \( 1 - 20 T + p^{3} T^{2} \)
19 \( 1 - 36 T + p^{3} T^{2} \)
29 \( 1 + 27 T + p^{3} T^{2} \)
31 \( 1 + 33 T + p^{3} T^{2} \)
37 \( 1 - 56 T + p^{3} T^{2} \)
41 \( 1 + 157 T + p^{3} T^{2} \)
43 \( 1 - 18 T + p^{3} T^{2} \)
47 \( 1 - 65 T + p^{3} T^{2} \)
53 \( 1 + 14 T + p^{3} T^{2} \)
59 \( 1 + 744 T + p^{3} T^{2} \)
61 \( 1 - 552 T + p^{3} T^{2} \)
67 \( 1 + 156 T + p^{3} T^{2} \)
71 \( 1 - 699 T + p^{3} T^{2} \)
73 \( 1 + 609 T + p^{3} T^{2} \)
79 \( 1 + 644 T + p^{3} T^{2} \)
83 \( 1 - 512 T + p^{3} T^{2} \)
89 \( 1 + 102 T + p^{3} T^{2} \)
97 \( 1 - 578 T + p^{3} T^{2} \)
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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

−11.50429916990015101305589071589, −10.39818643003529938431121028697, −9.414598983480301621781848422522, −8.107853789280623336959830261582, −7.19178629219405417593487885588, −5.96236358588767764811469126611, −4.95900924498192662948316038541, −3.46320835416819586200571433587, −2.58546977397398571614001380484, 0, 2.58546977397398571614001380484, 3.46320835416819586200571433587, 4.95900924498192662948316038541, 5.96236358588767764811469126611, 7.19178629219405417593487885588, 8.107853789280623336959830261582, 9.414598983480301621781848422522, 10.39818643003529938431121028697, 11.50429916990015101305589071589

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