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Properties

Label 2-133-133.102-c1-0-5
Degree $2$
Conductor $133$
Sign $-0.0416 - 0.999i$
Analytic cond. $1.06201$
Root an. cond. $1.03053$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.14 + 1.97i)2-s + 2.46·3-s + (−1.60 − 2.77i)4-s + (0.527 − 0.913i)5-s + (−2.81 + 4.86i)6-s + (1.23 + 2.33i)7-s + 2.74·8-s + 3.07·9-s + (1.20 + 2.08i)10-s + (0.0883 − 0.152i)11-s + (−3.94 − 6.83i)12-s + (−0.270 + 0.468i)13-s + (−6.03 − 0.226i)14-s + (1.29 − 2.25i)15-s + (0.0736 − 0.127i)16-s − 7.92·17-s + ⋯
L(s)  = 1  + (−0.806 + 1.39i)2-s + 1.42·3-s + (−0.800 − 1.38i)4-s + (0.235 − 0.408i)5-s + (−1.14 + 1.98i)6-s + (0.467 + 0.884i)7-s + 0.970·8-s + 1.02·9-s + (0.380 + 0.658i)10-s + (0.0266 − 0.0461i)11-s + (−1.13 − 1.97i)12-s + (−0.0750 + 0.130i)13-s + (−1.61 − 0.0606i)14-s + (0.335 − 0.581i)15-s + (0.0184 − 0.0319i)16-s − 1.92·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0416 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0416 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.0416 - 0.999i$
Analytic conductor: \(1.06201\)
Root analytic conductor: \(1.03053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 133,\ (\ :1/2),\ -0.0416 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.761646 + 0.794051i\)
\(L(\frac12)\) \(\approx\) \(0.761646 + 0.794051i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.23 - 2.33i)T \)
19 \( 1 + (-0.729 + 4.29i)T \)
good2 \( 1 + (1.14 - 1.97i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 2.46T + 3T^{2} \)
5 \( 1 + (-0.527 + 0.913i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.0883 + 0.152i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.270 - 0.468i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.92T + 17T^{2} \)
23 \( 1 + 1.47T + 23T^{2} \)
29 \( 1 + (-2.19 + 3.80i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.58 + 6.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.49 + 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.27 - 2.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.11 - 8.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.48T + 47T^{2} \)
53 \( 1 + (-0.716 - 1.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.933T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + (1.65 + 2.86i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.22 + 9.05i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.45T + 73T^{2} \)
79 \( 1 + (4.94 - 8.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.0234T + 83T^{2} \)
89 \( 1 - 3.93T + 89T^{2} \)
97 \( 1 + (-6.55 - 11.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.87797197314572256003515531674, −13.01018589200595821210767957567, −11.35343731858558977806061239767, −9.553833385124880003220297017029, −8.978866326683544737912338572353, −8.408480319777834748125570240318, −7.40237489887255213941655152945, −6.13950207545511351998702446384, −4.68782392012014183685411774876, −2.41176477324404986242070148205, 1.83559285462658029284416800332, 3.02693164854195143225574107862, 4.21842534287425453010921451939, 6.94159249693544758731272238639, 8.289671375156954825649406749629, 8.821228926028793930631266085109, 10.09447114985579232749928173943, 10.58637155046149980851794116778, 11.77968015148203796724399842289, 13.02102636575277518094462837593

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