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Properties

Label 2-3696-924.527-c0-0-1
Degree $2$
Conductor $3696$
Sign $-0.553 - 0.832i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
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 + i·7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.866 + 1.5i)17-s + (−0.866 + 1.5i)19-s + (−0.866 + 0.5i)21-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 0.999·27-s + 1.73·29-s + (0.499 − 0.866i)33-s + (−0.5 + 0.866i)37-s − 1.73·43-s + (−0.5 + 0.866i)47-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + i·7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.866 + 1.5i)17-s + (−0.866 + 1.5i)19-s + (−0.866 + 0.5i)21-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 0.999·27-s + 1.73·29-s + (0.499 − 0.866i)33-s + (−0.5 + 0.866i)37-s − 1.73·43-s + (−0.5 + 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.553 - 0.832i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3696} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3696,\ (\ :0),\ -0.553 - 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.300690582\)
\(L(\frac12)\) \(\approx\) \(1.300690582\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.73T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.73T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.607638404773675300498756173038, −8.360505739599596312663017057904, −8.041399836620591840986619587124, −6.43053108477506393518111422516, −5.97000029188545252031442503920, −5.17905538987832589625644209466, −4.34097057678247319323641695110, −3.43538748100146618661554237542, −2.77768371298720228872312493242, −1.75002477874755516307708948911, 0.68715686154561397740834493025, 1.87038119668336463679930121489, 2.85288805362582951474512305504, 3.58607698464127728205398289690, 4.75814704802173571329156375224, 5.30526924293699133434012409122, 6.74952027003886635987124694789, 6.92237760627241118935667185512, 7.62324586365053348350668194762, 8.263588305079994415976722656258

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