↑

Properties

Label 2-3696-3696.2309-c0-0-7
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

Related objects

Downloads

Learn more

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

\(L(\frac{1}{2})\) \(\approx\) \(1.276010841\)
\(L(\frac12)\) \(\approx\) \(1.276010841\)
\(L(1)\) 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 \( 1 - T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.847223904546809196778631868338, −7.41096598669066130912936897256, −6.74183960324660240834851292725, −6.25000201720753261695262757548, −5.02059322570927828845049396013, −4.59897832152653674075497237807, −4.08191702245062204688258928107, −3.13615877876061634596883600709, −1.96356863682943292043190655788, −0.65640797884856051454151002694, 1.46770742568001695622590945019, 2.81306828998642371766415608656, 3.39108194339758349849687032003, 4.83542512854989731225278815352, 5.34909034148314799722674917984, 5.83416919949955380414649956818, 6.67381008602171077392199544438, 7.22091131171543336356506192957, 8.144629971752926576316844321344, 8.495293102439822863459468611726

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