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Properties

Label 2-3648-456.125-c0-0-3
Degree $2$
Conductor $3648$
Sign $0.541 - 0.840i$
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  + (0.866 + 0.5i)3-s − 1.73·7-s + (0.499 + 0.866i)9-s + (1.5 − 0.866i)13-s + i·19-s + (−1.49 − 0.866i)21-s + (0.5 + 0.866i)25-s + 0.999i·27-s + 1.73·31-s + 1.73i·37-s + 1.73·39-s + (−0.866 − 0.5i)43-s + 1.99·49-s + (−0.5 + 0.866i)57-s + (−1.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s − 1.73·7-s + (0.499 + 0.866i)9-s + (1.5 − 0.866i)13-s + i·19-s + (−1.49 − 0.866i)21-s + (0.5 + 0.866i)25-s + 0.999i·27-s + 1.73·31-s + 1.73i·37-s + 1.73·39-s + (−0.866 − 0.5i)43-s + 1.99·49-s + (−0.5 + 0.866i)57-s + (−1.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $0.541 - 0.840i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ 0.541 - 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.523169727\)
\(L(\frac12)\) \(\approx\) \(1.523169727\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 - iT \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + 1.73T + T^{2} \)
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 \( 1 - 1.73T + T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.738091911340401417786731719244, −8.334894922464592906394213394383, −7.46619343142794961366432795992, −6.49250664103213528027518958080, −6.02696560172531877715583040914, −4.99625359641384536206022360759, −3.89427647525337997620399479887, −3.30522921439020997199549016748, −2.84017630922026782290377883631, −1.33194141698132996582936976828, 0.888606708636425191578682300979, 2.26199190579318636255818713658, 3.07566324530505089578852669065, 3.72642076446350956176807560447, 4.54659588479688318691082330447, 6.04242807978604945395513447565, 6.49063993620565985507147512727, 6.93615704810041792499628919411, 7.925543589113717373147911026400, 8.808647441171374201614369363160

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