Properties

Label 2-77658-1.1-c1-0-17
Degree $2$
Conductor $77658$
Sign $1$
Analytic cond. $620.102$
Root an. cond. $24.9018$
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  − 2-s + 3-s + 4-s + 2·5-s − 6-s + 7-s − 8-s + 9-s − 2·10-s − 4·11-s + 12-s + 6·13-s − 14-s + 2·15-s + 16-s + 2·17-s − 18-s + 4·19-s + 2·20-s + 21-s + 4·22-s + 8·23-s − 24-s − 25-s − 6·26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s + 1.66·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s + 0.852·22-s + 1.66·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77658 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77658 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77658\)    =    \(2 \cdot 3 \cdot 7 \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(620.102\)
Root analytic conductor: \(24.9018\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 77658,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.847811648\)
\(L(\frac12)\) \(\approx\) \(3.847811648\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
7 \( 1 - T \)
43 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.90904109514754, −13.53800432620863, −13.09566068443818, −12.80947476893668, −11.84608031639604, −11.45077635436338, −10.82518603097896, −10.51165699295953, −9.887509917660575, −9.556953998194817, −8.866526977701343, −8.538784136691204, −8.035516940773543, −7.451317409803031, −7.048382269445335, −6.206488582316572, −5.810617645550726, −5.266159387647769, −4.649645221352487, −3.768264319591088, −3.041954165773418, −2.743963951972667, −1.898891958756187, −1.289456507832505, −0.7567127900874061, 0.7567127900874061, 1.289456507832505, 1.898891958756187, 2.743963951972667, 3.041954165773418, 3.768264319591088, 4.649645221352487, 5.266159387647769, 5.810617645550726, 6.206488582316572, 7.048382269445335, 7.451317409803031, 8.035516940773543, 8.538784136691204, 8.866526977701343, 9.556953998194817, 9.887509917660575, 10.51165699295953, 10.82518603097896, 11.45077635436338, 11.84608031639604, 12.80947476893668, 13.09566068443818, 13.53800432620863, 13.90904109514754

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