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Properties

Label 2-34e2-1156.735-c0-0-0
Degree $2$
Conductor $1156$
Sign $-0.624 - 0.781i$
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  + (0.798 + 0.602i)2-s + (0.273 + 0.961i)4-s + (−0.739 + 1.67i)5-s + (−0.361 + 0.932i)8-s + (0.995 − 0.0922i)9-s + (−1.59 + 0.890i)10-s + (−1.85 − 0.719i)13-s + (−0.850 + 0.526i)16-s + (0.850 − 0.526i)17-s + (0.850 + 0.526i)18-s + (−1.81 − 0.252i)20-s + (−1.58 − 1.73i)25-s + (−1.04 − 1.69i)26-s + (1.11 + 0.621i)29-s + (−0.995 − 0.0922i)32-s + ⋯
L(s)  = 1  + (0.798 + 0.602i)2-s + (0.273 + 0.961i)4-s + (−0.739 + 1.67i)5-s + (−0.361 + 0.932i)8-s + (0.995 − 0.0922i)9-s + (−1.59 + 0.890i)10-s + (−1.85 − 0.719i)13-s + (−0.850 + 0.526i)16-s + (0.850 − 0.526i)17-s + (0.850 + 0.526i)18-s + (−1.81 − 0.252i)20-s + (−1.58 − 1.73i)25-s + (−1.04 − 1.69i)26-s + (1.11 + 0.621i)29-s + (−0.995 − 0.0922i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.624 - 0.781i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.624 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.402965593\)
\(L(\frac12)\) \(\approx\) \(1.402965593\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.798 - 0.602i)T \)
17 \( 1 + (-0.850 + 0.526i)T \)
good3 \( 1 + (-0.995 + 0.0922i)T^{2} \)
5 \( 1 + (0.739 - 1.67i)T + (-0.673 - 0.739i)T^{2} \)
7 \( 1 + (0.183 + 0.982i)T^{2} \)
11 \( 1 + (0.526 + 0.850i)T^{2} \)
13 \( 1 + (1.85 + 0.719i)T + (0.739 + 0.673i)T^{2} \)
19 \( 1 + (-0.273 + 0.961i)T^{2} \)
23 \( 1 + (0.183 + 0.982i)T^{2} \)
29 \( 1 + (-1.11 - 0.621i)T + (0.526 + 0.850i)T^{2} \)
31 \( 1 + (-0.673 + 0.739i)T^{2} \)
37 \( 1 + (-1.87 - 0.629i)T + (0.798 + 0.602i)T^{2} \)
41 \( 1 + (-0.0373 - 0.806i)T + (-0.995 + 0.0922i)T^{2} \)
43 \( 1 + (0.445 + 0.895i)T^{2} \)
47 \( 1 + (0.982 + 0.183i)T^{2} \)
53 \( 1 + (0.365 - 0.0339i)T + (0.982 - 0.183i)T^{2} \)
59 \( 1 + (0.932 + 0.361i)T^{2} \)
61 \( 1 + (-0.0762 - 0.0521i)T + (0.361 + 0.932i)T^{2} \)
67 \( 1 + (0.273 - 0.961i)T^{2} \)
71 \( 1 + (-0.183 - 0.982i)T^{2} \)
73 \( 1 + (0.352 - 1.49i)T + (-0.895 - 0.445i)T^{2} \)
79 \( 1 + (0.961 + 0.273i)T^{2} \)
83 \( 1 + (0.0922 - 0.995i)T^{2} \)
89 \( 1 + (-1.48 + 0.576i)T + (0.739 - 0.673i)T^{2} \)
97 \( 1 + (0.748 + 0.621i)T + (0.183 + 0.982i)T^{2} \)
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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

−10.21935499940888942025973771617, −9.762593977150344847138138418787, −8.096228351774222616595616232390, −7.52257914605923706546861241910, −7.04309841285125537248997669176, −6.31276947355514285771174942542, −5.08259477077727639052495317671, −4.26149583129492170160781036040, −3.15422621956112565266401166439, −2.61054270770460349266109043411, 1.04121157962032752093557134549, 2.25235674811074659467623072174, 3.80775828342829547855740950117, 4.60560529532195168929351880555, 4.90828957427837386733795194153, 6.08058663990796720446572543242, 7.33087650887909444780417822621, 7.929799132200622901843788299782, 9.229669862935233361047810481994, 9.650610946266755469408015830769

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