↑

Properties

Label 2-3696-924.335-c0-0-1
Degree $2$
Conductor $3696$
Sign $0.998 + 0.0457i$
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  + (−0.809 − 0.587i)3-s + (−0.587 + 1.80i)5-s + (0.809 − 0.587i)7-s + (0.309 + 0.951i)9-s + (0.951 − 0.309i)11-s + (1.53 − 1.11i)15-s + (0.363 − 1.11i)17-s + (−1.30 − 0.951i)19-s − 21-s + 1.17·23-s + (−2.11 − 1.53i)25-s + (0.309 − 0.951i)27-s + (0.190 + 0.587i)31-s + (−0.951 − 0.309i)33-s + (0.587 + 1.80i)35-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s + (−0.587 + 1.80i)5-s + (0.809 − 0.587i)7-s + (0.309 + 0.951i)9-s + (0.951 − 0.309i)11-s + (1.53 − 1.11i)15-s + (0.363 − 1.11i)17-s + (−1.30 − 0.951i)19-s − 21-s + 1.17·23-s + (−2.11 − 1.53i)25-s + (0.309 − 0.951i)27-s + (0.190 + 0.587i)31-s + (−0.951 − 0.309i)33-s + (0.587 + 1.80i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.998 + 0.0457i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3696} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3696,\ (\ :0),\ 0.998 + 0.0457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.008168330\)
\(L(\frac12)\) \(\approx\) \(1.008168330\)
\(L(1)\) 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 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.951 + 0.309i)T \)
good5 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 - 1.17T + T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + 1.17T + T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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

−8.471038279763947001157797415896, −7.60214347492742857547057092548, −7.14749048816856137407316744767, −6.65435465302682799203693952711, −6.01537190801208881141016732052, −4.82758303957473940886848759425, −4.18707215584046397038635735844, −3.10054147198535933506632363160, −2.27755675604320237148401169120, −0.892218431701685159497979849673, 1.00647549411877659162415949100, 1.83853770811376542077966872405, 3.73328773229342073417407774809, 4.32754227869055277955359000433, 4.76155117815539062638721931121, 5.70375925197119453153919849083, 6.08661190487915178277826690864, 7.34503466927004773784660031119, 8.241195828202142271937196769094, 8.732507111560711630023441801800

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