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Properties

Label 2-648-216.67-c0-0-0
Degree $2$
Conductor $648$
Sign $0.0581 + 0.998i$
Analytic cond. $0.323394$
Root an. cond. $0.568677$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.5 + 0.866i)8-s + (1.43 − 1.20i)11-s + (0.766 − 0.642i)16-s + (0.173 − 0.300i)17-s + (−0.766 − 1.32i)19-s + (−1.43 − 1.20i)22-s + (0.173 + 0.984i)25-s + (−0.766 − 0.642i)32-s + (−0.326 − 0.118i)34-s + (−1.17 + 0.984i)38-s + (−0.266 + 1.50i)41-s + (0.266 − 0.223i)43-s + (−0.939 + 1.62i)44-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.5 + 0.866i)8-s + (1.43 − 1.20i)11-s + (0.766 − 0.642i)16-s + (0.173 − 0.300i)17-s + (−0.766 − 1.32i)19-s + (−1.43 − 1.20i)22-s + (0.173 + 0.984i)25-s + (−0.766 − 0.642i)32-s + (−0.326 − 0.118i)34-s + (−1.17 + 0.984i)38-s + (−0.266 + 1.50i)41-s + (0.266 − 0.223i)43-s + (−0.939 + 1.62i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.0581 + 0.998i$
Analytic conductor: \(0.323394\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ 0.0581 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8217158000\)
\(L(\frac12)\) \(\approx\) \(0.8217158000\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 \)
good5 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 + 0.642i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
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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

−10.83643242999468063334365998221, −9.596819433714896461291783735135, −9.019821274546884449978466417902, −8.320586602233171739234683126954, −7.06791228490979353391985939948, −5.98450871464762345955013208741, −4.76728833577939548002084748431, −3.76314719529268763888071934575, −2.78836931095467073363034603674, −1.20756043082390257455401919427, 1.70823222980527210448508480564, 3.84692162109391219141052490840, 4.51387278780308344635790454175, 5.80841108431118560691542198890, 6.57248140176737376617591987982, 7.36450840741736240589288687058, 8.332982737380002827075095355233, 9.120983224877157611837960395433, 9.937274960844281281914244116190, 10.64345408863207523413592104101

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