Properties

Label 2-115-23.12-c1-0-2
Degree $2$
Conductor $115$
Sign $0.804 - 0.594i$
Analytic cond. $0.918279$
Root an. cond. $0.958269$
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  + (−1.30 + 0.835i)2-s + (0.142 − 0.989i)3-s + (0.162 − 0.355i)4-s + (0.959 − 0.281i)5-s + (0.642 + 1.40i)6-s + (1.57 + 1.81i)7-s + (−0.354 − 2.46i)8-s + (1.91 + 0.563i)9-s + (−1.01 + 1.16i)10-s + (3.19 + 2.05i)11-s + (−0.328 − 0.211i)12-s + (−1.07 + 1.23i)13-s + (−3.55 − 1.04i)14-s + (−0.142 − 0.989i)15-s + (3.03 + 3.49i)16-s + (−1.19 − 2.60i)17-s + ⋯
L(s)  = 1  + (−0.919 + 0.591i)2-s + (0.0821 − 0.571i)3-s + (0.0811 − 0.177i)4-s + (0.429 − 0.125i)5-s + (0.262 + 0.574i)6-s + (0.593 + 0.684i)7-s + (−0.125 − 0.870i)8-s + (0.639 + 0.187i)9-s + (−0.320 + 0.369i)10-s + (0.963 + 0.619i)11-s + (−0.0948 − 0.0609i)12-s + (−0.296 + 0.342i)13-s + (−0.950 − 0.279i)14-s + (−0.0367 − 0.255i)15-s + (0.757 + 0.874i)16-s + (−0.288 − 0.632i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.804 - 0.594i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ 0.804 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.736550 + 0.242754i\)
\(L(\frac12)\) \(\approx\) \(0.736550 + 0.242754i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (3.50 + 3.27i)T \)
good2 \( 1 + (1.30 - 0.835i)T + (0.830 - 1.81i)T^{2} \)
3 \( 1 + (-0.142 + 0.989i)T + (-2.87 - 0.845i)T^{2} \)
7 \( 1 + (-1.57 - 1.81i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (-3.19 - 2.05i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (1.07 - 1.23i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (1.19 + 2.60i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (0.754 - 1.65i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (3.57 + 7.83i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.185 - 1.28i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (6.58 + 1.93i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (6.01 - 1.76i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (1.72 - 11.9i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 - 4.66T + 47T^{2} \)
53 \( 1 + (-3.35 - 3.86i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (4.75 - 5.48i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (1.23 + 8.57i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-4.36 + 2.80i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (2.38 - 1.53i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (0.872 - 1.90i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-8.09 + 9.34i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (13.1 + 3.84i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-1.65 + 11.5i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-5.95 + 1.74i)T + (81.6 - 52.4i)T^{2} \)
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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

−13.69857327531907468357934183554, −12.56049229100915064158440580677, −11.80894335750320693566982193883, −10.07590803298307438123678700588, −9.254443132264763535425898100820, −8.239663147185042075623526392210, −7.23008356663628669120739544730, −6.26551571583723082422230471279, −4.43469900075623912426136651106, −1.85001603054626592117685022471, 1.56374161847528184207976961542, 3.78341123000989059494978050425, 5.33233567324296098300253193928, 7.02481889097418054872894896717, 8.503855053657814199497544800785, 9.371997401584068332655296130276, 10.36231616721328660712971500392, 10.90077102522599449095083239079, 12.09960963960549900371812002607, 13.61581580082426763266760005473

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