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Properties

Label 2-115-115.114-c2-0-18
Degree $2$
Conductor $115$
Sign $-0.198 + 0.980i$
Analytic cond. $3.13352$
Root an. cond. $1.77017$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.436i·2-s − 4.79i·3-s + 3.80·4-s + (−1.45 − 4.78i)5-s + 2.09·6-s − 5.38·7-s + 3.41i·8-s − 13.9·9-s + (2.09 − 0.633i)10-s + 12.9i·11-s − 18.2i·12-s − 9.52i·13-s − 2.35i·14-s + (−22.9 + 6.95i)15-s + 13.7·16-s + 19.7·17-s + ⋯
L(s)  = 1  + 0.218i·2-s − 1.59i·3-s + 0.952·4-s + (−0.290 − 0.956i)5-s + 0.349·6-s − 0.769·7-s + 0.426i·8-s − 1.55·9-s + (0.209 − 0.0633i)10-s + 1.18i·11-s − 1.52i·12-s − 0.732i·13-s − 0.167i·14-s + (−1.52 + 0.463i)15-s + 0.859·16-s + 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ -0.198 + 0.980i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.922354 - 1.12779i\)
\(L(\frac12)\) \(\approx\) \(0.922354 - 1.12779i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.45 + 4.78i)T \)
23 \( 1 + (-20.2 - 10.9i)T \)
good2 \( 1 - 0.436iT - 4T^{2} \)
3 \( 1 + 4.79iT - 9T^{2} \)
7 \( 1 + 5.38T + 49T^{2} \)
11 \( 1 - 12.9iT - 121T^{2} \)
13 \( 1 + 9.52iT - 169T^{2} \)
17 \( 1 - 19.7T + 289T^{2} \)
19 \( 1 + 27.9iT - 361T^{2} \)
29 \( 1 - 12.1T + 841T^{2} \)
31 \( 1 - 37.0T + 961T^{2} \)
37 \( 1 + 35.6T + 1.36e3T^{2} \)
41 \( 1 + 32.0T + 1.68e3T^{2} \)
43 \( 1 - 29.4T + 1.84e3T^{2} \)
47 \( 1 - 82.6iT - 2.20e3T^{2} \)
53 \( 1 - 11.4T + 2.80e3T^{2} \)
59 \( 1 - 19.5T + 3.48e3T^{2} \)
61 \( 1 + 21.8iT - 3.72e3T^{2} \)
67 \( 1 - 106.T + 4.48e3T^{2} \)
71 \( 1 - 13.5T + 5.04e3T^{2} \)
73 \( 1 + 7.93iT - 5.32e3T^{2} \)
79 \( 1 - 123. iT - 6.24e3T^{2} \)
83 \( 1 + 129.T + 6.88e3T^{2} \)
89 \( 1 - 49.8iT - 7.92e3T^{2} \)
97 \( 1 + 76.4T + 9.40e3T^{2} \)
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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

−12.56962915736506800601672590608, −12.51625369898680996678115332060, −11.36432786851205472092640682487, −9.765545038738999490596579585046, −8.260757399094942535043891310490, −7.36819441709321266234309114891, −6.63022581279852593443720839813, −5.29777825144423323261890739379, −2.79025813959523103057763876958, −1.14191815727195155233621034624, 3.06830243001115207901615147200, 3.71576151914709664135453473428, 5.71847202296072891433053613419, 6.78091179573878729140119311996, 8.379636439702992230439583621034, 9.935482756323725176056283317196, 10.35607599127224358100869230201, 11.30069435171404843803539080619, 12.13795615879758535185703226791, 14.00408159300393162159410590999

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