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Properties

Label 2-117-117.110-c1-0-9
Degree $2$
Conductor $117$
Sign $-0.418 + 0.908i$
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  + (0.642 − 2.39i)2-s + (−0.203 + 1.72i)3-s + (−3.61 − 2.08i)4-s + (0.943 − 3.52i)5-s + (3.99 + 1.59i)6-s + (0.770 + 0.770i)7-s + (−3.81 + 3.81i)8-s + (−2.91 − 0.699i)9-s + (−7.84 − 4.52i)10-s + (2.00 + 0.536i)11-s + (4.32 − 5.78i)12-s + (−1.75 + 3.15i)13-s + (2.34 − 1.35i)14-s + (5.86 + 2.33i)15-s + (2.52 + 4.37i)16-s + (2.40 + 4.17i)17-s + ⋯
L(s)  = 1  + (0.454 − 1.69i)2-s + (−0.117 + 0.993i)3-s + (−1.80 − 1.04i)4-s + (0.421 − 1.57i)5-s + (1.63 + 0.650i)6-s + (0.291 + 0.291i)7-s + (−1.34 + 1.34i)8-s + (−0.972 − 0.233i)9-s + (−2.47 − 1.43i)10-s + (0.604 + 0.161i)11-s + (1.24 − 1.67i)12-s + (−0.485 + 0.874i)13-s + (0.626 − 0.361i)14-s + (1.51 + 0.603i)15-s + (0.631 + 1.09i)16-s + (0.584 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.418 + 0.908i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.418 + 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.659383 - 1.03016i\)
\(L(\frac12)\) \(\approx\) \(0.659383 - 1.03016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.203 - 1.72i)T \)
13 \( 1 + (1.75 - 3.15i)T \)
good2 \( 1 + (-0.642 + 2.39i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-0.943 + 3.52i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.770 - 0.770i)T + 7iT^{2} \)
11 \( 1 + (-2.00 - 0.536i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.40 - 4.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.01 - 1.34i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.993T + 23T^{2} \)
29 \( 1 + (3.11 - 1.79i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.10 + 0.832i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.11 - 0.834i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.0790 - 0.0790i)T + 41iT^{2} \)
43 \( 1 - 9.22iT - 43T^{2} \)
47 \( 1 + (2.02 + 7.57i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 9.48iT - 53T^{2} \)
59 \( 1 + (0.146 + 0.545i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 8.12T + 61T^{2} \)
67 \( 1 + (3.90 - 3.90i)T - 67iT^{2} \)
71 \( 1 + (-0.773 + 2.88i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.89 - 4.89i)T + 73iT^{2} \)
79 \( 1 + (5.66 - 9.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.38 + 0.638i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.0437 + 0.163i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.94 + 4.94i)T - 97iT^{2} \)
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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

−12.85360109334030590466897979183, −12.03436553218771069700680266130, −11.38604111890173697158999932376, −9.986301288611057840862786600717, −9.418698403536338869360698901196, −8.586724719191353233474499445328, −5.53089377461559499195581484358, −4.75844563818108617428503199214, −3.69423354434887751730192475994, −1.63713404100926908373283467251, 3.13192938719404329123681524628, 5.33546344248782927460994651793, 6.25796865426140108429247199468, 7.36876382811951931877566725570, 7.54031642221331661153075790502, 9.268526766418915889393901251176, 10.83529025732418860653167122813, 12.08758607842605487390933424328, 13.46763834624480990206056000535, 14.08912869145523497067831595166

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