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Properties

Label 2-612-68.67-c0-0-0
Degree $2$
Conductor $612$
Sign $1$
Analytic cond. $0.305427$
Root an. cond. $0.552655$
Motivic weight $0$
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 + 4-s + 8-s − 2·13-s + 16-s − 17-s + 25-s − 2·26-s + 32-s − 34-s − 49-s + 50-s − 2·52-s − 2·53-s + 64-s − 68-s + 2·89-s − 98-s + 100-s + 2·101-s − 2·104-s − 2·106-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s − 2·13-s + 16-s − 17-s + 25-s − 2·26-s + 32-s − 34-s − 49-s + 50-s − 2·52-s − 2·53-s + 64-s − 68-s + 2·89-s − 98-s + 100-s + 2·101-s − 2·104-s − 2·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(612\)    =    \(2^{2} \cdot 3^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(0.305427\)
Root analytic conductor: \(0.552655\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{612} (271, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 612,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.569616600\)
\(L(\frac12)\) \(\approx\) \(1.569616600\)
\(L(1)\) 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 \( 1 \)
17 \( 1 + T \)
good5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( ( 1 + T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 + T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−11.01361511296499662988051015116, −10.15934166306212921952565676537, −9.230068915079004819121392749342, −7.929900660265677048069781859922, −7.10461071371140693459800334853, −6.33572911611919867273565186629, −5.05357000362313766271212661191, −4.54815223255620235231881143477, −3.12333612951879445036966965769, −2.11496417031398947575557591105, 2.11496417031398947575557591105, 3.12333612951879445036966965769, 4.54815223255620235231881143477, 5.05357000362313766271212661191, 6.33572911611919867273565186629, 7.10461071371140693459800334853, 7.929900660265677048069781859922, 9.230068915079004819121392749342, 10.15934166306212921952565676537, 11.01361511296499662988051015116

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