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Properties

Label 2-117-117.61-c1-0-10
Degree $2$
Conductor $117$
Sign $-0.708 + 0.706i$
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  + (−0.567 − 0.983i)2-s + (−0.529 − 1.64i)3-s + (0.354 − 0.614i)4-s + (−0.0587 − 0.101i)5-s + (−1.32 + 1.45i)6-s + 0.849·7-s − 3.07·8-s + (−2.43 + 1.74i)9-s + (−0.0667 + 0.115i)10-s + (0.231 + 0.401i)11-s + (−1.20 − 0.259i)12-s + (1.54 − 3.25i)13-s + (−0.482 − 0.835i)14-s + (−0.136 + 0.150i)15-s + (1.03 + 1.79i)16-s + (−1.26 − 2.18i)17-s + ⋯
L(s)  = 1  + (−0.401 − 0.695i)2-s + (−0.305 − 0.952i)3-s + (0.177 − 0.307i)4-s + (−0.0262 − 0.0454i)5-s + (−0.539 + 0.595i)6-s + 0.321·7-s − 1.08·8-s + (−0.813 + 0.582i)9-s + (−0.0210 + 0.0365i)10-s + (0.0699 + 0.121i)11-s + (−0.346 − 0.0749i)12-s + (0.427 − 0.903i)13-s + (−0.128 − 0.223i)14-s + (−0.0352 + 0.0389i)15-s + (0.259 + 0.449i)16-s + (−0.306 − 0.530i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.708 + 0.706i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.708 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.309932 - 0.749675i\)
\(L(\frac12)\) \(\approx\) \(0.309932 - 0.749675i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.529 + 1.64i)T \)
13 \( 1 + (-1.54 + 3.25i)T \)
good2 \( 1 + (0.567 + 0.983i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.0587 + 0.101i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 0.849T + 7T^{2} \)
11 \( 1 + (-0.231 - 0.401i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.26 + 2.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.09 - 5.35i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.65T + 23T^{2} \)
29 \( 1 + (0.952 + 1.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.657 - 1.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.01 - 3.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.68T + 41T^{2} \)
43 \( 1 + 4.20T + 43T^{2} \)
47 \( 1 + (-1.34 + 2.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.389T + 53T^{2} \)
59 \( 1 + (5.53 - 9.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 7.76T + 61T^{2} \)
67 \( 1 + 1.02T + 67T^{2} \)
71 \( 1 + (-3.61 - 6.25i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.31T + 73T^{2} \)
79 \( 1 + (4.41 - 7.64i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.75 + 3.04i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.62 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−12.85584412866220725730857877945, −11.99034568874310555469036635849, −11.13966605491124937877786430787, −10.26550085105633411757407983506, −8.900815151425785073921589684991, −7.74424268227618969438017825339, −6.42826779536506570856092290412, −5.31486929170476192376201013789, −2.85133477626061990091243335229, −1.18485209533362802415571065924, 3.25703540749123049062106340358, 4.81716467107423941203225762731, 6.22278224490399775763056534334, 7.32551636632228325904099187167, 8.812676975038615818993178074839, 9.289964140726441735943400669077, 11.00328817151713506194802731534, 11.47244929267208526115123465421, 12.85295003634411925583082825652, 14.29460139467314303050378507910

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