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Properties

Label 2-3696-3696.2309-c0-0-10
Degree $2$
Conductor $3696$
Sign $-0.382 + 0.923i$
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  + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + 1.00i·6-s − i·7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (−0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (−1 + i)13-s + (0.707 + 0.707i)14-s − 1.00·16-s + (0.707 + 0.707i)18-s + (1 − i)19-s + (−0.707 − 0.707i)21-s + 1.00·22-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + 1.00i·6-s − i·7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (−0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (−1 + i)13-s + (0.707 + 0.707i)14-s − 1.00·16-s + (0.707 + 0.707i)18-s + (1 − i)19-s + (−0.707 − 0.707i)21-s + 1.00·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3696} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3696,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7886570797\)
\(L(\frac12)\) \(\approx\) \(0.7886570797\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + iT \)
11 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + iT^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-1 + i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1 + i)T - iT^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.394746797372499003900357274229, −7.53434386109938538704011423547, −7.21973278813186634488805335180, −6.69039857964553564530477812456, −5.64022633051779227425812840371, −4.83927425243526971299563440970, −3.79596689472966908656620266538, −2.68201806813300423558113843435, −1.71580070647071614491177051355, −0.49608840477102763492301179845, 1.79520814450877843079955475644, 2.56890252297312000046964673872, 3.19314376998177353432153886406, 4.10146448574086131006820655679, 5.15803561652222012755930574629, 5.61929077008526890949319899612, 7.26449505944484082453238310108, 7.70306881845802960930203541215, 8.286062609035717701092674136532, 9.104743692162840015991582229119

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