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Properties

Label 2-34e2-68.67-c0-0-1
Degree $2$
Conductor $1156$
Sign $0.727 + 0.685i$
Analytic cond. $0.576919$
Root an. cond. $0.759551$
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 − 1.41i·5-s + 8-s − 9-s − 1.41i·10-s + 16-s − 18-s − 1.41i·20-s − 1.00·25-s + 1.41i·29-s + 32-s − 36-s + 1.41i·37-s − 1.41i·40-s − 1.41i·41-s + ⋯
L(s)  = 1  + 2-s + 4-s − 1.41i·5-s + 8-s − 9-s − 1.41i·10-s + 16-s − 18-s − 1.41i·20-s − 1.00·25-s + 1.41i·29-s + 32-s − 36-s + 1.41i·37-s − 1.41i·40-s − 1.41i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.727 + 0.685i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (1155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.727 + 0.685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.846842273\)
\(L(\frac12)\) \(\approx\) \(1.846842273\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.959446504008254711568932284596, −8.847343217806489063264158348341, −8.372049316369935133831786348577, −7.32600854090312147157857780631, −6.28213338680234465645477220488, −5.37658945327439208851325846734, −4.91609366761207145121749398379, −3.86530813933462331101078733180, −2.78962838836108292014508091453, −1.41958524975628106160777037168, 2.20994499824788278960940488802, 2.98841318741524794401284343643, 3.79083432070679022376672819701, 4.98648292657619979258461325750, 6.06216260690761447288166444172, 6.44387746314973525475619203676, 7.46786936584930805246552501014, 8.123104844868391067328305122397, 9.473336792913580224820822284835, 10.37782133896535897058727791420

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