Properties

Label 2-15680-1.1-c1-0-31
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 + 2·11-s + 4·13-s + 15-s + 6·19-s + 3·23-s + 25-s + 5·27-s + 3·29-s − 2·33-s + 12·37-s − 4·39-s + 7·41-s + 9·43-s + 2·45-s + 6·53-s − 2·55-s − 6·57-s − 10·59-s + 5·61-s − 4·65-s − 11·67-s − 3·69-s − 10·71-s + 8·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.603·11-s + 1.10·13-s + 0.258·15-s + 1.37·19-s + 0.625·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 0.348·33-s + 1.97·37-s − 0.640·39-s + 1.09·41-s + 1.37·43-s + 0.298·45-s + 0.824·53-s − 0.269·55-s − 0.794·57-s − 1.30·59-s + 0.640·61-s − 0.496·65-s − 1.34·67-s − 0.361·69-s − 1.18·71-s + 0.936·73-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

\(L(1)\) \(\approx\) \(1.930481768\)
\(L(\frac12)\) \(\approx\) \(1.930481768\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 17 T + p T^{2} \)
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

−16.04630752123941, −15.60899303073951, −14.83098623691399, −14.35847813010524, −13.79922324029524, −13.23690526311393, −12.52213325992671, −11.93303886203286, −11.51814369428591, −10.98287936813612, −10.64091741710147, −9.600254345793332, −9.189969868172366, −8.554212621869222, −7.897145348461680, −7.306176459786158, −6.555625793091157, −5.915990565476152, −5.559739820464971, −4.594790261054093, −4.070146875333812, −3.182161854066597, −2.653674206690180, −1.238683050159451, −0.7329365122804462, 0.7329365122804462, 1.238683050159451, 2.653674206690180, 3.182161854066597, 4.070146875333812, 4.594790261054093, 5.559739820464971, 5.915990565476152, 6.555625793091157, 7.306176459786158, 7.897145348461680, 8.554212621869222, 9.189969868172366, 9.600254345793332, 10.64091741710147, 10.98287936813612, 11.51814369428591, 11.93303886203286, 12.52213325992671, 13.23690526311393, 13.79922324029524, 14.35847813010524, 14.83098623691399, 15.60899303073951, 16.04630752123941

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