Properties

Label 2-3640-3640.1357-c0-0-1
Degree $2$
Conductor $3640$
Sign $0.661 - 0.749i$
Analytic cond. $1.81659$
Root an. cond. $1.34781$
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  + 2-s + 4-s + (−0.707 − 0.707i)5-s + i·7-s + 8-s + i·9-s + (−0.707 − 0.707i)10-s + (−0.707 + 0.707i)13-s + i·14-s + 16-s + i·18-s + (1.41 + 1.41i)19-s + (−0.707 − 0.707i)20-s + (−1 − i)23-s + 1.00i·25-s + (−0.707 + 0.707i)26-s + ⋯
L(s)  = 1  + 2-s + 4-s + (−0.707 − 0.707i)5-s + i·7-s + 8-s + i·9-s + (−0.707 − 0.707i)10-s + (−0.707 + 0.707i)13-s + i·14-s + 16-s + i·18-s + (1.41 + 1.41i)19-s + (−0.707 − 0.707i)20-s + (−1 − i)23-s + 1.00i·25-s + (−0.707 + 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (1357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ 0.661 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.135885683\)
\(L(\frac12)\) \(\approx\) \(2.135885683\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 - iT \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 - iT^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1 + i)T - iT^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + iT^{2} \)
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

−8.585154755341563990106239624428, −7.924077542552287779973685778957, −7.45381032857878614910756251525, −6.41395999092546758011073058369, −5.54836700878340647908528228622, −5.05961432111678777325759964453, −4.34485062934813516604677373640, −3.49313771771896681891951132144, −2.43609213270872656410953348102, −1.66232066301673123775747521073, 0.910706297014820153260356641748, 2.52943712198888682254080657087, 3.37200176047433054491382579043, 3.80926429032578453526854788226, 4.72219311410426133692717007630, 5.54395367591771906483163770696, 6.48136347046059531427752747270, 7.20733534094975036539282178375, 7.42810941086616045529009085760, 8.320218487681190207632552834164

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