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Properties

Label 2-115-5.4-c3-0-13
Degree $2$
Conductor $115$
Sign $0.830 + 0.556i$
Analytic cond. $6.78521$
Root an. cond. $2.60484$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.00i·2-s + 1.85i·3-s − 1.00·4-s + (−6.22 + 9.28i)5-s + 5.55·6-s + 4.63i·7-s − 20.9i·8-s + 23.5·9-s + (27.8 + 18.6i)10-s + 70.0·11-s − 1.86i·12-s − 4.48i·13-s + 13.9·14-s + (−17.2 − 11.5i)15-s − 71.0·16-s + 62.4i·17-s + ⋯
L(s)  = 1  − 1.06i·2-s + 0.356i·3-s − 0.126·4-s + (−0.556 + 0.830i)5-s + 0.378·6-s + 0.250i·7-s − 0.927i·8-s + 0.872·9-s + (0.881 + 0.590i)10-s + 1.92·11-s − 0.0449i·12-s − 0.0956i·13-s + 0.265·14-s + (−0.296 − 0.198i)15-s − 1.11·16-s + 0.890i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.830 + 0.556i$
Analytic conductor: \(6.78521\)
Root analytic conductor: \(2.60484\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3/2),\ 0.830 + 0.556i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.74716 - 0.530986i\)
\(L(\frac12)\) \(\approx\) \(1.74716 - 0.530986i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (6.22 - 9.28i)T \)
23 \( 1 + 23iT \)
good2 \( 1 + 3.00iT - 8T^{2} \)
3 \( 1 - 1.85iT - 27T^{2} \)
7 \( 1 - 4.63iT - 343T^{2} \)
11 \( 1 - 70.0T + 1.33e3T^{2} \)
13 \( 1 + 4.48iT - 2.19e3T^{2} \)
17 \( 1 - 62.4iT - 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
29 \( 1 + 1.08T + 2.43e4T^{2} \)
31 \( 1 + 129.T + 2.97e4T^{2} \)
37 \( 1 - 13.3iT - 5.06e4T^{2} \)
41 \( 1 + 298.T + 6.89e4T^{2} \)
43 \( 1 + 120. iT - 7.95e4T^{2} \)
47 \( 1 - 542. iT - 1.03e5T^{2} \)
53 \( 1 - 78.0iT - 1.48e5T^{2} \)
59 \( 1 + 12.3T + 2.05e5T^{2} \)
61 \( 1 + 81.5T + 2.26e5T^{2} \)
67 \( 1 + 1.00e3iT - 3.00e5T^{2} \)
71 \( 1 - 201.T + 3.57e5T^{2} \)
73 \( 1 + 520. iT - 3.89e5T^{2} \)
79 \( 1 + 1.10e3T + 4.93e5T^{2} \)
83 \( 1 + 227. iT - 5.71e5T^{2} \)
89 \( 1 + 727.T + 7.04e5T^{2} \)
97 \( 1 + 1.01e3iT - 9.12e5T^{2} \)
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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

−12.53695884801806764424192497154, −11.82746527221970980102940680234, −10.99840016109359583948246347678, −10.05592932540092281532578018078, −9.146708779150080155100575957869, −7.35682342098428035472112608545, −6.39685223838818216245046476962, −4.15439334310159932916294689576, −3.34176534936445397495314766380, −1.49787357281580763163738971861, 1.30181772185760300942581149489, 3.98854525063382701289053349681, 5.32087582716795169515969208017, 6.84077457633095674365236935923, 7.37084080176675109292258451383, 8.647759396566278283119760702655, 9.624635626337643836841382512162, 11.53979209652799461644349084093, 12.01497894581249481782979929359, 13.41350087381243021501270792190

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