Properties

Label 2-3696-3696.2309-c0-0-5
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

\(L(\frac{1}{2})\) \(\approx\) \(1.487852121\)
\(L(\frac12)\) \(\approx\) \(1.487852121\)
\(L(1)\) 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

−8.737032344243545006942887830057, −7.933928074606190766041037431268, −7.23471009172685199965502751420, −6.10425999936415416901769798122, −5.43940154537612672429560153011, −5.03700578812925240563420007747, −4.33327176608740484043463720650, −3.48459442119935604399606584580, −2.72988614739859624462827227150, −0.821630196187036966879707990220, 1.14608094220888338400354180758, 2.37811722475327698297844180189, 3.37787020493153626111754490817, 3.97742715288329574186804021405, 4.85425537220580614167125952494, 5.68376533320453529160530179190, 6.69087385311231333438315573621, 7.02727857803045116178634325173, 7.62175040108114883151529984919, 8.114151394976741554441188702036

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