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Properties

Label 2-3640-3640.3037-c0-0-2
Degree $2$
Conductor $3640$
Sign $-0.256 - 0.966i$
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  + i·2-s − 4-s + (0.707 − 0.707i)5-s + 7-s − i·8-s + i·9-s + (0.707 + 0.707i)10-s + (−0.707 + 0.707i)13-s + i·14-s + 16-s − 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  + i·2-s − 4-s + (0.707 − 0.707i)5-s + 7-s − i·8-s + i·9-s + (0.707 + 0.707i)10-s + (−0.707 + 0.707i)13-s + i·14-s + 16-s − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $-0.256 - 0.966i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (3037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ -0.256 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.347669574\)
\(L(\frac12)\) \(\approx\) \(1.347669574\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 - T \)
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 \( 1 - 1.41T + T^{2} \)
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

−8.657578490760869676469604823873, −8.223079130101030943584281105446, −7.54092580829686917424553556458, −6.75392143812410845579809843959, −5.80531525292778865673820589110, −5.21834861936545400972303261861, −4.66673008758376070602786812834, −3.93845347110604291189880482406, −2.24662548278131948506723686353, −1.44885226837640473930033497283, 0.839370229332453173968129738486, 2.18719344386117848685370072742, 2.68516667309838955539178839105, 3.68358328648679010955504681732, 4.70037423891867523129908703349, 5.21387994567582464179321901966, 6.27629819720536208937120242244, 6.96482988920740425703211835218, 7.936522663462771561098275446537, 8.823787920532041927444989212747

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