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Properties

Label 2-133-133.10-c1-0-10
Degree $2$
Conductor $133$
Sign $-0.611 - 0.791i$
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  + (−1.52 − 1.81i)2-s + (−0.439 − 2.49i)3-s + (−0.623 + 3.53i)4-s + (−1.04 + 0.184i)5-s + (−3.84 + 4.58i)6-s + (−0.591 − 2.57i)7-s + (3.26 − 1.88i)8-s + (−3.18 + 1.16i)9-s + (1.92 + 1.61i)10-s − 2.17·11-s + 9.08·12-s + (4.74 + 3.98i)13-s + (−3.77 + 4.99i)14-s + (0.917 + 2.52i)15-s + (−1.61 − 0.588i)16-s + (1.08 − 2.97i)17-s + ⋯
L(s)  = 1  + (−1.07 − 1.28i)2-s + (−0.253 − 1.43i)3-s + (−0.311 + 1.76i)4-s + (−0.467 + 0.0824i)5-s + (−1.56 + 1.87i)6-s + (−0.223 − 0.974i)7-s + (1.15 − 0.665i)8-s + (−1.06 + 0.386i)9-s + (0.607 + 0.510i)10-s − 0.656·11-s + 2.62·12-s + (1.31 + 1.10i)13-s + (−1.00 + 1.33i)14-s + (0.236 + 0.651i)15-s + (−0.404 − 0.147i)16-s + (0.262 − 0.720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.611 - 0.791i$
Analytic conductor: \(1.06201\)
Root analytic conductor: \(1.03053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 133,\ (\ :1/2),\ -0.611 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.178087 + 0.362561i\)
\(L(\frac12)\) \(\approx\) \(0.178087 + 0.362561i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.591 + 2.57i)T \)
19 \( 1 + (2.57 + 3.51i)T \)
good2 \( 1 + (1.52 + 1.81i)T + (-0.347 + 1.96i)T^{2} \)
3 \( 1 + (0.439 + 2.49i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (1.04 - 0.184i)T + (4.69 - 1.71i)T^{2} \)
11 \( 1 + 2.17T + 11T^{2} \)
13 \( 1 + (-4.74 - 3.98i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.08 + 2.97i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (4.58 + 3.84i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.85 + 0.327i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-0.545 - 0.945i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.26 + 3.61i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.49 + 3.77i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (4.46 + 1.62i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.63 - 4.48i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (1.86 + 0.328i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-9.40 - 3.42i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-6.67 + 7.95i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-9.19 + 10.9i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-3.22 + 8.87i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-3.68 + 0.648i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.99 - 5.49i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-8.54 - 4.93i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.14 - 6.49i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (2.13 + 12.0i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.39753482781848467973095521780, −11.39044764605321757590170824751, −10.90051923011729591487613246808, −9.579499000262112365531195387059, −8.312821888951373586056730318014, −7.51773178210447787674590425405, −6.43795045213455444485276896242, −3.88729773565505927305122091253, −2.15381244638495666907614637675, −0.61152147060681640547688238642, 3.75037833464762693024362985515, 5.51339711492188987594581211275, 6.03072503249940143700816017122, 8.027175690369944214823895031200, 8.466862230928425251872017583149, 9.744115546772427045789734490221, 10.30187101214909844389931742984, 11.46357037280792581933053584087, 12.98495806458507061541735826527, 14.74574867841517582383671159273

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