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Properties

Label 2-117-117.2-c1-0-4
Degree $2$
Conductor $117$
Sign $-0.491 + 0.871i$
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  + (−1.90 − 1.90i)2-s + (1.70 − 0.320i)3-s + 5.28i·4-s + (0.518 − 1.93i)5-s + (−3.86 − 2.63i)6-s + (0.463 − 1.73i)7-s + (6.27 − 6.27i)8-s + (2.79 − 1.09i)9-s + (−4.68 + 2.70i)10-s + (−2.74 + 2.74i)11-s + (1.69 + 8.99i)12-s + (−1.27 − 3.37i)13-s + (−4.18 + 2.41i)14-s + (0.261 − 3.46i)15-s − 13.3·16-s + (−0.899 + 1.55i)17-s + ⋯
L(s)  = 1  + (−1.34 − 1.34i)2-s + (0.982 − 0.185i)3-s + 2.64i·4-s + (0.232 − 0.865i)5-s + (−1.57 − 1.07i)6-s + (0.175 − 0.654i)7-s + (2.21 − 2.21i)8-s + (0.931 − 0.364i)9-s + (−1.48 + 0.855i)10-s + (−0.826 + 0.826i)11-s + (0.489 + 2.59i)12-s + (−0.353 − 0.935i)13-s + (−1.11 + 0.646i)14-s + (0.0675 − 0.893i)15-s − 3.34·16-s + (−0.218 + 0.377i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.491 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.491 + 0.871i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.491 + 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.383562 - 0.656512i\)
\(L(\frac12)\) \(\approx\) \(0.383562 - 0.656512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.320i)T \)
13 \( 1 + (1.27 + 3.37i)T \)
good2 \( 1 + (1.90 + 1.90i)T + 2iT^{2} \)
5 \( 1 + (-0.518 + 1.93i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.463 + 1.73i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.74 - 2.74i)T - 11iT^{2} \)
17 \( 1 + (0.899 - 1.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.15 - 4.32i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.01 + 1.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.08iT - 29T^{2} \)
31 \( 1 + (-2.37 - 0.635i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.695 - 2.59i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.78 + 0.478i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.43 - 0.829i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.69 - 6.33i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 10.2iT - 53T^{2} \)
59 \( 1 + (7.05 - 7.05i)T - 59iT^{2} \)
61 \( 1 + (5.91 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.19 - 8.19i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.19 + 1.39i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (10.1 + 10.1i)T + 73iT^{2} \)
79 \( 1 + (1.00 - 1.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.20 + 1.66i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (4.60 + 1.23i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (14.1 + 3.78i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.69699646905278382038373425438, −12.41463009802059677703556590394, −10.64232060188013020455288653692, −10.02373337282208887431303549965, −9.068617987038402866233310863057, −8.072685481768809741260376416313, −7.45632652406455679611583877986, −4.42272431321644653506953597354, −2.88555022472228452964788005292, −1.39683229176145883145948089766, 2.44749887843628076239514886118, 5.14815508460130725623439938365, 6.59335856265524821211206673723, 7.46891727696090954927523252468, 8.552160550897359046139178657507, 9.270972759482761272871210142702, 10.23993289429023595050910539508, 11.24775012426259356782663240403, 13.55979431027308764994252484870, 14.25261078824758510112589953025

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