Properties

Label 2-10304-1.1-c1-0-33
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 + 6·17-s − 6·19-s − 2·21-s + 23-s − 5·25-s − 4·27-s − 10·29-s − 4·31-s + 8·33-s + 2·37-s − 10·41-s − 4·43-s − 12·47-s + 49-s + 12·51-s + 6·53-s − 12·57-s − 2·59-s − 63-s + 2·69-s + 8·71-s − 6·73-s − 10·75-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.45·17-s − 1.37·19-s − 0.436·21-s + 0.208·23-s − 25-s − 0.769·27-s − 1.85·29-s − 0.718·31-s + 1.39·33-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 1.68·51-s + 0.824·53-s − 1.58·57-s − 0.260·59-s − 0.125·63-s + 0.240·69-s + 0.949·71-s − 0.702·73-s − 1.15·75-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 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.82932825791711, −16.53627012674020, −15.48630555173004, −14.93841290373823, −14.69942615469793, −14.12081664854446, −13.50528016643797, −12.94575810266352, −12.41935582126153, −11.55782353717944, −11.26379153059736, −10.12418726450732, −9.794271656572296, −9.148093868475362, −8.632941414197224, −8.037277722996302, −7.405673891886393, −6.708379789577481, −5.980249573980565, −5.330108524546335, −4.200030024740632, −3.602095908503061, −3.227972568938517, −2.079132253846368, −1.551615123121714, 0, 1.551615123121714, 2.079132253846368, 3.227972568938517, 3.602095908503061, 4.200030024740632, 5.330108524546335, 5.980249573980565, 6.708379789577481, 7.405673891886393, 8.037277722996302, 8.632941414197224, 9.148093868475362, 9.794271656572296, 10.12418726450732, 11.26379153059736, 11.55782353717944, 12.41935582126153, 12.94575810266352, 13.50528016643797, 14.12081664854446, 14.69942615469793, 14.93841290373823, 15.48630555173004, 16.53627012674020, 16.82932825791711

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