Properties

Label 2-77315-1.1-c1-0-2
Degree $2$
Conductor $77315$
Sign $1$
Analytic cond. $617.363$
Root an. cond. $24.8467$
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 − 2·4-s + 5-s + 7-s − 2·9-s + 3·11-s − 2·12-s − 5·13-s + 15-s + 4·16-s + 3·17-s − 2·19-s − 2·20-s + 21-s + 6·23-s + 25-s − 5·27-s − 2·28-s − 3·29-s + 4·31-s + 3·33-s + 35-s + 4·36-s + 2·37-s − 5·39-s + 12·41-s + 10·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s + 0.727·17-s − 0.458·19-s − 0.447·20-s + 0.218·21-s + 1.25·23-s + 1/5·25-s − 0.962·27-s − 0.377·28-s − 0.557·29-s + 0.718·31-s + 0.522·33-s + 0.169·35-s + 2/3·36-s + 0.328·37-s − 0.800·39-s + 1.87·41-s + 1.52·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77315\)    =    \(5 \cdot 7 \cdot 47^{2}\)
Sign: $1$
Analytic conductor: \(617.363\)
Root analytic conductor: \(24.8467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{77315} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 77315,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.894748881\)
\(L(\frac12)\) \(\approx\) \(2.894748881\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 - T \)
47 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.14903506400788, −13.65808613708156, −13.10036819282079, −12.57869943230203, −12.21100072338902, −11.56324467134773, −11.03934615279457, −10.41607678691302, −9.758756457102174, −9.465426109532644, −8.961331924744946, −8.657177493926324, −7.932205738893120, −7.527347396250062, −6.965048783006685, −6.122391304952919, −5.590517987923631, −5.189757953955260, −4.466985612093940, −4.051209707319724, −3.349279849093354, −2.613918311366194, −2.235700998928465, −1.145498296175759, −0.6177865421306052, 0.6177865421306052, 1.145498296175759, 2.235700998928465, 2.613918311366194, 3.349279849093354, 4.051209707319724, 4.466985612093940, 5.189757953955260, 5.590517987923631, 6.122391304952919, 6.965048783006685, 7.527347396250062, 7.932205738893120, 8.657177493926324, 8.961331924744946, 9.465426109532644, 9.758756457102174, 10.41607678691302, 11.03934615279457, 11.56324467134773, 12.21100072338902, 12.57869943230203, 13.10036819282079, 13.65808613708156, 14.14903506400788

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