Properties

Label 2-15680-1.1-c1-0-50
Degree $2$
Conductor $15680$
Sign $1$
Analytic cond. $125.205$
Root an. cond. $11.1895$
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  + 3-s − 5-s − 2·9-s + 5·11-s + 7·13-s − 15-s + 3·17-s − 2·19-s + 8·23-s + 25-s − 5·27-s + 5·29-s + 10·31-s + 5·33-s − 4·37-s + 7·39-s + 6·41-s − 2·43-s + 2·45-s + 7·47-s + 3·51-s + 10·53-s − 5·55-s − 2·57-s − 10·59-s − 12·61-s − 7·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 2/3·9-s + 1.50·11-s + 1.94·13-s − 0.258·15-s + 0.727·17-s − 0.458·19-s + 1.66·23-s + 1/5·25-s − 0.962·27-s + 0.928·29-s + 1.79·31-s + 0.870·33-s − 0.657·37-s + 1.12·39-s + 0.937·41-s − 0.304·43-s + 0.298·45-s + 1.02·47-s + 0.420·51-s + 1.37·53-s − 0.674·55-s − 0.264·57-s − 1.30·59-s − 1.53·61-s − 0.868·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15680\)    =    \(2^{6} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(125.205\)
Root analytic conductor: \(11.1895\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{15680} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.459594375\)
\(L(\frac12)\) \(\approx\) \(3.459594375\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−15.89104627706304, −15.33944797543052, −14.94477292641748, −14.24505121442230, −13.82677049992335, −13.45345176960504, −12.62461496177551, −11.88251493548986, −11.69043887495879, −10.80734656654518, −10.59938766360557, −9.485671416399019, −8.984058911318888, −8.597783593714082, −8.143428776875302, −7.347696585229453, −6.494057934027413, −6.239870637926173, −5.404769925065928, −4.432013891326752, −3.878021031863203, −3.246158722888102, −2.691479284900285, −1.376430557304477, −0.9066465586783946, 0.9066465586783946, 1.376430557304477, 2.691479284900285, 3.246158722888102, 3.878021031863203, 4.432013891326752, 5.404769925065928, 6.239870637926173, 6.494057934027413, 7.347696585229453, 8.143428776875302, 8.597783593714082, 8.984058911318888, 9.485671416399019, 10.59938766360557, 10.80734656654518, 11.69043887495879, 11.88251493548986, 12.62461496177551, 13.45345176960504, 13.82677049992335, 14.24505121442230, 14.94477292641748, 15.33944797543052, 15.89104627706304

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