Properties

Label 2-10304-1.1-c1-0-16
Degree $2$
Conductor $10304$
Sign $-1$
Analytic cond. $82.2778$
Root an. cond. $9.07071$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 7-s + 9-s + 4·11-s − 2·13-s − 4·19-s + 2·21-s + 23-s − 5·25-s + 4·27-s + 6·29-s + 2·31-s − 8·33-s + 2·37-s + 4·39-s − 6·41-s + 4·43-s − 10·47-s + 49-s + 6·53-s + 8·57-s − 6·59-s − 4·61-s − 63-s + 4·67-s − 2·69-s + 8·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.917·19-s + 0.436·21-s + 0.208·23-s − 25-s + 0.769·27-s + 1.11·29-s + 0.359·31-s − 1.39·33-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s − 1.45·47-s + 1/7·49-s + 0.824·53-s + 1.05·57-s − 0.781·59-s − 0.512·61-s − 0.125·63-s + 0.488·67-s − 0.240·69-s + 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10304\)    =    \(2^{6} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(82.2778\)
Root analytic conductor: \(9.07071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10304} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−17.00012728642254, −16.53300415043381, −15.81176377458403, −15.28097094971232, −14.56470235119990, −14.08811085024020, −13.36478700889056, −12.66846617951719, −12.08958132119920, −11.76384385417602, −11.17644250657891, −10.48263395584682, −9.961574268699181, −9.311957318432471, −8.626621031318346, −7.931194250136415, −7.005831832554492, −6.419420437226026, −6.168875738235536, −5.259664942734222, −4.624579710215873, −3.954606202784269, −3.047022400590179, −2.038864973499305, −0.9891462512697895, 0, 0.9891462512697895, 2.038864973499305, 3.047022400590179, 3.954606202784269, 4.624579710215873, 5.259664942734222, 6.168875738235536, 6.419420437226026, 7.005831832554492, 7.931194250136415, 8.626621031318346, 9.311957318432471, 9.961574268699181, 10.48263395584682, 11.17644250657891, 11.76384385417602, 12.08958132119920, 12.66846617951719, 13.36478700889056, 14.08811085024020, 14.56470235119990, 15.28097094971232, 15.81176377458403, 16.53300415043381, 17.00012728642254

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