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Properties

Label 2-117-117.41-c1-0-4
Degree $2$
Conductor $117$
Sign $0.382 - 0.923i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
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  + (0.0826 − 0.0221i)2-s + (0.405 + 1.68i)3-s + (−1.72 + 0.996i)4-s + (1.73 − 0.466i)5-s + (0.0707 + 0.130i)6-s + (0.362 + 0.362i)7-s + (−0.241 + 0.241i)8-s + (−2.67 + 1.36i)9-s + (0.133 − 0.0770i)10-s + (1.48 + 5.53i)11-s + (−2.37 − 2.50i)12-s + (−0.119 − 3.60i)13-s + (0.0379 + 0.0219i)14-s + (1.48 + 2.74i)15-s + (1.97 − 3.42i)16-s + (2.73 − 4.74i)17-s + ⋯
L(s)  = 1  + (0.0584 − 0.0156i)2-s + (0.233 + 0.972i)3-s + (−0.862 + 0.498i)4-s + (0.777 − 0.208i)5-s + (0.0288 + 0.0531i)6-s + (0.136 + 0.136i)7-s + (−0.0853 + 0.0853i)8-s + (−0.890 + 0.454i)9-s + (0.0421 − 0.0243i)10-s + (0.447 + 1.66i)11-s + (−0.686 − 0.722i)12-s + (−0.0332 − 0.999i)13-s + (0.0101 + 0.00585i)14-s + (0.384 + 0.707i)15-s + (0.494 − 0.856i)16-s + (0.663 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.899619 + 0.601013i\)
\(L(\frac12)\) \(\approx\) \(0.899619 + 0.601013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.405 - 1.68i)T \)
13 \( 1 + (0.119 + 3.60i)T \)
good2 \( 1 + (-0.0826 + 0.0221i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.73 + 0.466i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.362 - 0.362i)T + 7iT^{2} \)
11 \( 1 + (-1.48 - 5.53i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.73 + 4.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.92 + 7.17i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 4.43T + 23T^{2} \)
29 \( 1 + (-0.446 - 0.257i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.854 - 3.18i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.815 - 3.04i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.97 + 1.97i)T + 41iT^{2} \)
43 \( 1 - 2.09iT - 43T^{2} \)
47 \( 1 + (4.06 + 1.08i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 - 8.79iT - 53T^{2} \)
59 \( 1 + (-7.47 - 2.00i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 5.79T + 61T^{2} \)
67 \( 1 + (3.78 - 3.78i)T - 67iT^{2} \)
71 \( 1 + (2.49 - 0.668i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.21 + 4.21i)T + 73iT^{2} \)
79 \( 1 + (2.43 + 4.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.879 - 3.28i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (10.8 + 2.90i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.56 - 1.56i)T - 97iT^{2} \)
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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

−13.70812977730793009421331694727, −12.88881226204245830541152717905, −11.73175082916765137708568108868, −10.20652234201906845179385271069, −9.488217687072945090883411026209, −8.754218936093945266443235422101, −7.30481933082032617143508969696, −5.25513389688950504716651635709, −4.61417131389974173991234271356, −2.91010402478407962891029983903, 1.51597348080417927992314190880, 3.70218679983023759125284803709, 5.77158690682178030871000058272, 6.31608307121196184058266203225, 8.124195627414708169028494960772, 8.918161076508812338040249609590, 10.09667248374199745858143004971, 11.31342988667296218104894409960, 12.60501789449499822439418377412, 13.54809398293390575699594912117

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