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Properties

Label 2-1040-1040.779-c0-0-0
Degree $2$
Conductor $1040$
Sign $0.382 - 0.923i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 + 0.707i)5-s − 1.41·7-s + (0.707 − 0.707i)8-s + i·9-s − 1.00i·10-s + (−0.707 + 0.707i)13-s + (1.00 + 1.00i)14-s − 1.00·16-s + (0.707 − 0.707i)18-s + (−0.707 + 0.707i)20-s + 1.00i·25-s + 1.00·26-s − 1.41i·28-s + (−1 + i)29-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 + 0.707i)5-s − 1.41·7-s + (0.707 − 0.707i)8-s + i·9-s − 1.00i·10-s + (−0.707 + 0.707i)13-s + (1.00 + 1.00i)14-s − 1.00·16-s + (0.707 − 0.707i)18-s + (−0.707 + 0.707i)20-s + 1.00i·25-s + 1.00·26-s − 1.41i·28-s + (−1 + i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :0),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5586534663\)
\(L(\frac12)\) \(\approx\) \(0.5586534663\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 - iT^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (1 - i)T - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1 + i)T - iT^{2} \)
67 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.41iT - T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−10.23512411425828770105837169660, −9.534324021491830308132952037606, −9.053453945066551871355325838224, −7.75417776661479623000932467562, −7.03980812798886009311897710015, −6.31256852057017075490478618277, −5.04612917643598632778801314456, −3.68916834089050664158142784518, −2.77191509882949237773878583823, −1.93197752641356831956654957775, 0.62699861284878271219952904396, 2.34075259030305639607332404767, 3.75349347066729923678781924202, 5.13493022132377374504014551198, 5.92736534421073017342400873849, 6.53805972561543607744218087568, 7.40350341651876617857951604536, 8.500478771782120723669583431592, 9.225077755970592861906893649576, 9.857246868427324304909699876252

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