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Properties

Label 2-32674-1.1-c1-0-0
Degree $2$
Conductor $32674$
Sign $1$
Analytic cond. $260.903$
Root an. cond. $16.1524$
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 + 2·3-s + 4-s + 2·6-s − 4·7-s + 8-s + 9-s − 6·11-s + 2·12-s − 2·13-s − 4·14-s + 16-s + 17-s + 18-s − 4·19-s − 8·21-s − 6·22-s + 2·24-s − 5·25-s − 2·26-s − 4·27-s − 4·28-s + 32-s − 12·33-s + 34-s + 36-s + 4·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.577·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s − 1.74·21-s − 1.27·22-s + 0.408·24-s − 25-s − 0.392·26-s − 0.769·27-s − 0.755·28-s + 0.176·32-s − 2.08·33-s + 0.171·34-s + 1/6·36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32674 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32674 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32674\)    =    \(2 \cdot 17 \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(260.903\)
Root analytic conductor: \(16.1524\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32674,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.008894345\)
\(L(\frac12)\) \(\approx\) \(2.008894345\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−14.90982094777104, −14.58991348929425, −13.87098835422677, −13.31693692672158, −13.09225738672901, −12.74013829735905, −12.05893929506502, −11.41284414948318, −10.59647448652075, −10.18811178374536, −9.689885249549889, −9.205115072853645, −8.356973566260885, −7.995505358896280, −7.397286555311264, −6.845500567012502, −6.065703713234614, −5.639589494999747, −4.928192906439477, −4.100160317974860, −3.554355138597574, −2.897144070443922, −2.530659310123555, −1.988024378752425, −0.3989783898679184, 0.3989783898679184, 1.988024378752425, 2.530659310123555, 2.897144070443922, 3.554355138597574, 4.100160317974860, 4.928192906439477, 5.639589494999747, 6.065703713234614, 6.845500567012502, 7.397286555311264, 7.995505358896280, 8.356973566260885, 9.205115072853645, 9.689885249549889, 10.18811178374536, 10.59647448652075, 11.41284414948318, 12.05893929506502, 12.74013829735905, 13.09225738672901, 13.31693692672158, 13.87098835422677, 14.58991348929425, 14.90982094777104

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