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Properties

Label 2-3648-228.107-c0-0-0
Degree $2$
Conductor $3648$
Sign $-0.813 - 0.582i$
Analytic cond. $1.82058$
Root an. cond. $1.34929$
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  + (0.5 + 0.866i)3-s + 1.73i·7-s + (−0.499 + 0.866i)9-s + (−1.5 − 0.866i)13-s + 19-s + (−1.49 + 0.866i)21-s + (−0.5 + 0.866i)25-s − 0.999·27-s + 31-s + 1.73i·37-s − 1.73i·39-s + (−1.5 + 0.866i)43-s − 1.99·49-s + (0.5 + 0.866i)57-s + (0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + 1.73i·7-s + (−0.499 + 0.866i)9-s + (−1.5 − 0.866i)13-s + 19-s + (−1.49 + 0.866i)21-s + (−0.5 + 0.866i)25-s − 0.999·27-s + 31-s + 1.73i·37-s − 1.73i·39-s + (−1.5 + 0.866i)43-s − 1.99·49-s + (0.5 + 0.866i)57-s + (0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $-0.813 - 0.582i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (3071, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ -0.813 - 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188523118\)
\(L(\frac12)\) \(\approx\) \(1.188523118\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 - T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 - 1.73iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.164396358342400192339265504917, −8.135856214364261383347781877329, −8.004748633111801118525207755015, −6.78439128378872207816584944527, −5.73318233333090770682897061127, −5.16157004050832001179611148795, −4.70115629763185022505170522789, −3.21858897364455392257496573221, −2.89639698451218657079293897813, −1.93211352640927179597508577210, 0.61281377287679957099647736819, 1.78700753699350091902442170232, 2.73756909422922703652687762421, 3.78198779750635217529127922722, 4.41377226684315555116930439258, 5.42673723889931766938193549615, 6.58361091504117671013423817651, 7.08117446889672850619941354036, 7.52884280340215885232502880889, 8.193700708612186487124219598072

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