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Properties

Label 2-32276-1.1-c1-0-1
Degree $2$
Conductor $32276$
Sign $-1$
Analytic cond. $257.725$
Root an. cond. $16.0538$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $3$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·5-s − 4·7-s + 9-s − 6·11-s − 5·13-s + 6·15-s − 2·17-s + 8·21-s − 6·23-s + 4·25-s + 4·27-s − 5·29-s − 8·31-s + 12·33-s + 12·35-s − 6·37-s + 10·39-s − 11·41-s − 2·43-s − 3·45-s + 47-s + 9·49-s + 4·51-s − 10·53-s + 18·55-s + 3·59-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s − 1.51·7-s + 1/3·9-s − 1.80·11-s − 1.38·13-s + 1.54·15-s − 0.485·17-s + 1.74·21-s − 1.25·23-s + 4/5·25-s + 0.769·27-s − 0.928·29-s − 1.43·31-s + 2.08·33-s + 2.02·35-s − 0.986·37-s + 1.60·39-s − 1.71·41-s − 0.304·43-s − 0.447·45-s + 0.145·47-s + 9/7·49-s + 0.560·51-s − 1.37·53-s + 2.42·55-s + 0.390·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32276\)    =    \(2^{2} \cdot 8069\)
Sign: $-1$
Analytic conductor: \(257.725\)
Root analytic conductor: \(16.0538\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((2,\ 32276,\ (\ :1/2),\ -1)\)

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 \( 1 \)
8069 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−15.86582080591028, −15.46686264437905, −15.10663326234044, −14.31338039773539, −13.51259761626943, −12.95948107010232, −12.60412392105708, −12.01366628706534, −11.84512005011585, −10.97198204528576, −10.62505229525659, −10.09452176681509, −9.596271227546942, −8.814560014228252, −8.097449800211904, −7.511292041774654, −7.159676922215664, −6.518002299967063, −5.825419397525468, −5.273039738500142, −4.804138105007842, −4.022565008701552, −3.305225793484217, −2.806044090839094, −1.882759535213278, 0, 0, 0, 1.882759535213278, 2.806044090839094, 3.305225793484217, 4.022565008701552, 4.804138105007842, 5.273039738500142, 5.825419397525468, 6.518002299967063, 7.159676922215664, 7.511292041774654, 8.097449800211904, 8.814560014228252, 9.596271227546942, 10.09452176681509, 10.62505229525659, 10.97198204528576, 11.84512005011585, 12.01366628706534, 12.60412392105708, 12.95948107010232, 13.51259761626943, 14.31338039773539, 15.10663326234044, 15.46686264437905, 15.86582080591028

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