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Properties

Label 2-1044-116.115-c0-0-1
Degree $2$
Conductor $1044$
Sign $-i$
Analytic cond. $0.521023$
Root an. cond. $0.721819$
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 − i·8-s + 2·13-s + 16-s + 2i·17-s − 25-s + 2i·26-s + i·29-s + i·32-s − 2·34-s − 2i·41-s + 49-s − i·50-s − 2·52-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·8-s + 2·13-s + 16-s + 2i·17-s − 25-s + 2i·26-s + i·29-s + i·32-s − 2·34-s − 2i·41-s + 49-s − i·50-s − 2·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $-i$
Analytic conductor: \(0.521023\)
Root analytic conductor: \(0.721819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9735082239\)
\(L(\frac12)\) \(\approx\) \(0.9735082239\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
29 \( 1 - iT \)
good5 \( 1 + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 2T + T^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 2iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 2iT - T^{2} \)
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

−10.39314412286586381959102286234, −9.136582950307374215712904953706, −8.566846727412326940650735572456, −7.943770739202514330549570501359, −6.86923401100398445214128329178, −6.04520953621082194752600492984, −5.54305916696474736197309645428, −4.07755343616041517777261341117, −3.61958708754889052001443921957, −1.52402115992100717149398770874, 1.11510224376861701012205912788, 2.53025936657014135303776606490, 3.53932277484929359482759347776, 4.42507415835623693882361471806, 5.47853995808691265790792018113, 6.37443722451672287028766051737, 7.67819777817118561754574462403, 8.451676580302565599692990197680, 9.301359604656973864110643321350, 9.881028017428466162534318982973

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