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Properties

Label 2-117-117.59-c1-0-7
Degree $2$
Conductor $117$
Sign $0.984 - 0.173i$
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  + (0.311 − 0.311i)2-s + (1.53 + 0.809i)3-s + 1.80i·4-s + (−0.789 − 2.94i)5-s + (0.729 − 0.224i)6-s + (0.0751 + 0.280i)7-s + (1.18 + 1.18i)8-s + (1.68 + 2.47i)9-s + (−1.16 − 0.671i)10-s + (−1.04 − 1.04i)11-s + (−1.46 + 2.76i)12-s + (−3.58 − 0.362i)13-s + (0.110 + 0.0639i)14-s + (1.17 − 5.14i)15-s − 2.87·16-s + (0.767 + 1.32i)17-s + ⋯
L(s)  = 1  + (0.220 − 0.220i)2-s + (0.884 + 0.467i)3-s + 0.902i·4-s + (−0.352 − 1.31i)5-s + (0.297 − 0.0917i)6-s + (0.0284 + 0.106i)7-s + (0.419 + 0.419i)8-s + (0.563 + 0.826i)9-s + (−0.367 − 0.212i)10-s + (−0.316 − 0.316i)11-s + (−0.422 + 0.798i)12-s + (−0.994 − 0.100i)13-s + (0.0296 + 0.0170i)14-s + (0.303 − 1.32i)15-s − 0.718·16-s + (0.186 + 0.322i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38054 + 0.120541i\)
\(L(\frac12)\) \(\approx\) \(1.38054 + 0.120541i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.53 - 0.809i)T \)
13 \( 1 + (3.58 + 0.362i)T \)
good2 \( 1 + (-0.311 + 0.311i)T - 2iT^{2} \)
5 \( 1 + (0.789 + 2.94i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.0751 - 0.280i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.04 + 1.04i)T + 11iT^{2} \)
17 \( 1 + (-0.767 - 1.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.989 + 3.69i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.92 + 6.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.80iT - 29T^{2} \)
31 \( 1 + (3.94 - 1.05i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.40 - 8.97i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.37 + 0.636i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.95 - 5.16i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.30 + 8.58i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 - 8.42iT - 53T^{2} \)
59 \( 1 + (-8.77 - 8.77i)T + 59iT^{2} \)
61 \( 1 + (-3.29 + 5.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.95 - 7.28i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (4.14 + 1.11i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-4.56 + 4.56i)T - 73iT^{2} \)
79 \( 1 + (2.96 + 5.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.4 + 3.07i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (1.37 - 0.367i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.26 + 0.874i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.33675629931404492521413020628, −12.66696187895202277521345313235, −11.80932627410440907029727425533, −10.34065304502800930308860479322, −8.975878119276239338249095124219, −8.384837530354157857301099400165, −7.41929409445412038842931547570, −5.00784692986418132793111549501, −4.14556669020168338689504649882, −2.62167191824847110212356302328, 2.26772852025973535844186357646, 3.87118651219387596835552445318, 5.76088233996956687008911564942, 7.14323387763798027139412667816, 7.60704886589443642612733730083, 9.482086974319521463507012763948, 10.18971644698547945049570712524, 11.39291991324261792104977460819, 12.66171930373513034237706036805, 14.03414228539307212385823320099

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