Properties

Label 2-3696-3696.2309-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.3846003222\)
\(L(\frac12)\) \(\approx\) \(0.3846003222\)
\(L(1)\) 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

−8.774163689661369185423440592102, −8.165433317139602992155868064586, −7.28666197069968562097276048366, −6.82417320872108960930365000574, −5.73769980910223247850783044723, −4.66162652594714173620312044567, −3.93826934619315182441673716256, −3.73807585504115601458433168742, −1.78937204770319098185640147088, −0.73920905033876603328891847385, 0.53012419786571848095327523369, 2.23128279064347330851027903244, 2.75850407653702511671254820587, 3.99125758522670388942875904521, 5.27199729035796345586345788002, 6.05520081442777269604329021064, 6.75230925054840958158873410696, 7.06391350212354678381919213998, 7.997153866278728595547961213562, 8.436271010862283010868977961889

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