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Properties

Label 2-1040-1040.259-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: $\chi_{1040} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.478810281\)
\(L(\frac12)\) \(\approx\) \(1.478810281\)
\(L(1)\) 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

−10.22599389611638223906200185872, −9.257212732592679157935367351049, −8.383800961912270556035506241003, −7.36300456147231296634852935355, −6.45947604000961988501938732943, −5.59367200220802434548998603140, −4.32002358656696978796945152672, −3.91118033309518516630462113715, −2.68980435341209283407962396602, −1.39650715523926000449103022997, 1.80398406833872364908613193021, 3.38727970396561220770471484071, 4.41020746037465488779234338140, 5.09998631041694403050545089909, 5.66881651040936602670631241031, 7.13934297188828263054490812346, 7.83719368044416238617072164309, 8.322435289836134849541167417871, 8.971152740236731555063579123132, 10.57528943162120993551082102831

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