Properties

Label 2-2394-1.1-c3-0-97
Degree $2$
Conductor $2394$
Sign $-1$
Analytic cond. $141.250$
Root an. cond. $11.8848$
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 + 4·4-s + 10·5-s + 7·7-s − 8·8-s − 20·10-s − 8·11-s − 50·13-s − 14·14-s + 16·16-s − 114·17-s + 19·19-s + 40·20-s + 16·22-s + 148·23-s − 25·25-s + 100·26-s + 28·28-s + 30·29-s + 304·31-s − 32·32-s + 228·34-s + 70·35-s − 274·37-s − 38·38-s − 80·40-s + 202·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s − 0.219·11-s − 1.06·13-s − 0.267·14-s + 1/4·16-s − 1.62·17-s + 0.229·19-s + 0.447·20-s + 0.155·22-s + 1.34·23-s − 1/5·25-s + 0.754·26-s + 0.188·28-s + 0.192·29-s + 1.76·31-s − 0.176·32-s + 1.15·34-s + 0.338·35-s − 1.21·37-s − 0.162·38-s − 0.316·40-s + 0.769·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-1$
Analytic conductor: \(141.250\)
Root analytic conductor: \(11.8848\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2394} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2394,\ (\ :3/2),\ -1)\)

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 \( 1 + p T \)
3 \( 1 \)
7 \( 1 - p T \)
19 \( 1 - p T \)
good5 \( 1 - 2 p T + p^{3} T^{2} \)
11 \( 1 + 8 T + p^{3} T^{2} \)
13 \( 1 + 50 T + p^{3} T^{2} \)
17 \( 1 + 114 T + p^{3} T^{2} \)
23 \( 1 - 148 T + p^{3} T^{2} \)
29 \( 1 - 30 T + p^{3} T^{2} \)
31 \( 1 - 304 T + p^{3} T^{2} \)
37 \( 1 + 274 T + p^{3} T^{2} \)
41 \( 1 - 202 T + p^{3} T^{2} \)
43 \( 1 + 116 T + p^{3} T^{2} \)
47 \( 1 - 324 T + p^{3} T^{2} \)
53 \( 1 - 550 T + p^{3} T^{2} \)
59 \( 1 + 628 T + p^{3} T^{2} \)
61 \( 1 + 58 T + p^{3} T^{2} \)
67 \( 1 + 756 T + p^{3} T^{2} \)
71 \( 1 - 216 T + p^{3} T^{2} \)
73 \( 1 + 278 T + p^{3} T^{2} \)
79 \( 1 + 952 T + p^{3} T^{2} \)
83 \( 1 - 1184 T + p^{3} T^{2} \)
89 \( 1 + 1542 T + p^{3} T^{2} \)
97 \( 1 + 870 T + p^{3} T^{2} \)
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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

−8.425398285888150257616978913904, −7.40399186802543261518996593762, −6.85268757751299862731943155820, −6.00581360934126391753699133931, −5.10862375676951993458250424912, −4.38245245971149070948713368972, −2.82657944644151904778975881842, −2.26067686156543308061415249168, −1.23445907955584138301360236049, 0, 1.23445907955584138301360236049, 2.26067686156543308061415249168, 2.82657944644151904778975881842, 4.38245245971149070948713368972, 5.10862375676951993458250424912, 6.00581360934126391753699133931, 6.85268757751299862731943155820, 7.40399186802543261518996593762, 8.425398285888150257616978913904

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