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Properties

Label 2-117-117.25-c1-0-7
Degree $2$
Conductor $117$
Sign $0.689 + 0.724i$
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  + (−0.784 − 0.452i)2-s + (1.66 + 0.460i)3-s + (−0.589 − 1.02i)4-s + (1.94 − 1.12i)5-s + (−1.10 − 1.11i)6-s + (−2.97 − 1.71i)7-s + 2.87i·8-s + (2.57 + 1.53i)9-s − 2.03·10-s + (3.20 + 1.84i)11-s + (−0.514 − 1.97i)12-s + (0.351 − 3.58i)13-s + (1.55 + 2.69i)14-s + (3.76 − 0.977i)15-s + (0.124 − 0.214i)16-s − 4.21·17-s + ⋯
L(s)  = 1  + (−0.554 − 0.320i)2-s + (0.963 + 0.265i)3-s + (−0.294 − 0.510i)4-s + (0.868 − 0.501i)5-s + (−0.449 − 0.456i)6-s + (−1.12 − 0.649i)7-s + 1.01i·8-s + (0.858 + 0.512i)9-s − 0.642·10-s + (0.965 + 0.557i)11-s + (−0.148 − 0.570i)12-s + (0.0975 − 0.995i)13-s + (0.415 + 0.720i)14-s + (0.970 − 0.252i)15-s + (0.0310 − 0.0537i)16-s − 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.689 + 0.724i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.689 + 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.957267 - 0.410698i\)
\(L(\frac12)\) \(\approx\) \(0.957267 - 0.410698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 - 0.460i)T \)
13 \( 1 + (-0.351 + 3.58i)T \)
good2 \( 1 + (0.784 + 0.452i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.94 + 1.12i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.97 + 1.71i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.20 - 1.84i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 4.21T + 17T^{2} \)
19 \( 1 - 4.25iT - 19T^{2} \)
23 \( 1 + (-1.89 - 3.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.18 - 2.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.37 - 3.67i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.49iT - 37T^{2} \)
41 \( 1 + (6.86 - 3.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.450 - 0.779i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.80 + 2.77i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.59T + 53T^{2} \)
59 \( 1 + (-4.44 + 2.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.50 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.7 + 6.75i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.65iT - 71T^{2} \)
73 \( 1 + 5.45iT - 73T^{2} \)
79 \( 1 + (5.46 - 9.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.465 - 0.268i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.75iT - 89T^{2} \)
97 \( 1 + (5.87 + 3.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.40143230107982246341748762757, −12.76336373259862910330285331826, −10.83121685063368374694325914066, −9.780258316377928525007366455403, −9.504686665092798056131391408041, −8.454034912647726025969702301497, −6.86455718668087813120121539190, −5.32599373527335684059814418127, −3.67914876013593911457956688986, −1.72295543123498472015543982165, 2.53697279939410730977316605982, 3.90817296472796277278895598477, 6.46349799337274418705150121467, 6.91617479351308405201288242627, 8.738993521100839561617763979971, 9.097951473642473871499542452683, 9.952728676839495909420975267732, 11.73319637879754192067246102983, 13.07426304107862626787839918356, 13.49107156545091226207821703312

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