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Properties

Label 2-2132-2132.735-c0-0-0
Degree $2$
Conductor $2132$
Sign $0.175 + 0.984i$
Analytic cond. $1.06400$
Root an. cond. $1.03150$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 1.93i·5-s + 0.999·8-s + (0.258 − 0.965i)9-s + (1.67 + 0.965i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.158 − 0.207i)17-s + (0.707 + 0.707i)18-s + (−1.67 + 0.965i)20-s − 2.73·25-s + (0.499 + 0.866i)26-s + (−0.965 + 0.741i)29-s + (−0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 1.93i·5-s + 0.999·8-s + (0.258 − 0.965i)9-s + (1.67 + 0.965i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.158 − 0.207i)17-s + (0.707 + 0.707i)18-s + (−1.67 + 0.965i)20-s − 2.73·25-s + (0.499 + 0.866i)26-s + (−0.965 + 0.741i)29-s + (−0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2132\)    =    \(2^{2} \cdot 13 \cdot 41\)
Sign: $0.175 + 0.984i$
Analytic conductor: \(1.06400\)
Root analytic conductor: \(1.03150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2132} (735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2132,\ (\ :0),\ 0.175 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8051992098\)
\(L(\frac12)\) \(\approx\) \(0.8051992098\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + T \)
good3 \( 1 + (-0.258 + 0.965i)T^{2} \)
5 \( 1 + 1.93iT - T^{2} \)
7 \( 1 + (-0.258 - 0.965i)T^{2} \)
11 \( 1 + (-0.965 - 0.258i)T^{2} \)
17 \( 1 + (-0.158 + 0.207i)T + (-0.258 - 0.965i)T^{2} \)
19 \( 1 + (0.965 - 0.258i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.965 - 0.741i)T + (0.258 - 0.965i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.707 + 0.707i)T^{2} \)
53 \( 1 + (-0.465 + 1.12i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.258 + 0.965i)T^{2} \)
71 \( 1 + (0.258 + 0.965i)T^{2} \)
73 \( 1 + 0.517iT - T^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.0999 + 0.758i)T + (-0.965 - 0.258i)T^{2} \)
97 \( 1 + (1.83 - 0.241i)T + (0.965 - 0.258i)T^{2} \)
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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

−8.889610655854574711899454697784, −8.468156320256842799437525569808, −7.75230540249550513111314007058, −6.80743192403410646774914275100, −5.81985617707516132763676991504, −5.31332619853648334899732359309, −4.49618485459504062060226365398, −3.60086273358793823374368510076, −1.57293371712309118599950351884, −0.70967855210929748315508724350, 1.90468907873951712107003342662, 2.47996048657346224774204091044, 3.56428690408387412406703378057, 4.14654008615343933687086005075, 5.50246042164293650594386867321, 6.58976812311533515698422965659, 7.27040187164591912584893018251, 7.84348672963832017897027181339, 8.754842005360814493287977169226, 9.799597419747595531777588329376

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