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Properties

Label 2-3696-3696.2309-c0-0-6
Degree $2$
Conductor $3696$
Sign $0.991 - 0.130i$
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  + (−0.965 − 0.258i)2-s + (0.707 − 0.707i)3-s + (0.866 + 0.499i)4-s + (1.22 + 1.22i)5-s + (−0.866 + 0.500i)6-s + i·7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−0.866 − 1.49i)10-s + (0.707 + 0.707i)11-s + (0.965 − 0.258i)12-s + (0.366 − 0.366i)13-s + (0.258 − 0.965i)14-s + 1.73·15-s + (0.500 + 0.866i)16-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.707 − 0.707i)3-s + (0.866 + 0.499i)4-s + (1.22 + 1.22i)5-s + (−0.866 + 0.500i)6-s + i·7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−0.866 − 1.49i)10-s + (0.707 + 0.707i)11-s + (0.965 − 0.258i)12-s + (0.366 − 0.366i)13-s + (0.258 − 0.965i)14-s + 1.73·15-s + (0.500 + 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.991 - 0.130i$
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: \((2,\ 3696,\ (\ :0),\ 0.991 - 0.130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.457508235\)
\(L(\frac12)\) \(\approx\) \(1.457508235\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.707 - 0.707i)T \)
good5 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
13 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + 1.93T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
61 \( 1 + (1 - i)T - iT^{2} \)
67 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT - T^{2} \)
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

−8.972808360117259080394155499683, −7.988669287736579817337569898189, −7.24601500431398705610846314133, −6.66247077592510988404143130158, −6.19996401068151252651244565534, −5.23028226852783312841754547019, −3.37023777292671164636793250101, −2.96813286484832490510466634777, −2.10105118604150912259887840830, −1.51094149433640210047622881630, 1.24310203167974656171765794664, 1.75189228189111757248876285884, 3.15212757531964451354729328010, 4.03503416120053825541573565232, 5.05351269783690979120263664973, 5.74773604673331922592600263322, 6.44842012152287878985315083288, 7.53609843746398531107449215565, 8.197514342899504838913562873795, 8.753470363898230911394626019915

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