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Properties

Label 2-115-115.114-c4-0-34
Degree $2$
Conductor $115$
Sign $-0.785 + 0.618i$
Analytic cond. $11.8875$
Root an. cond. $3.44783$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.94i·2-s − 6.82i·3-s − 8.45·4-s + (22.1 + 11.6i)5-s − 33.7·6-s + 47.4·7-s − 37.2i·8-s + 34.3·9-s + (57.5 − 109. i)10-s − 121. i·11-s + 57.7i·12-s − 25.2i·13-s − 234. i·14-s + (79.4 − 151. i)15-s − 319.·16-s + 130.·17-s + ⋯
L(s)  = 1  − 1.23i·2-s − 0.758i·3-s − 0.528·4-s + (0.884 + 0.465i)5-s − 0.938·6-s + 0.967·7-s − 0.582i·8-s + 0.424·9-s + (0.575 − 1.09i)10-s − 1.00i·11-s + 0.401i·12-s − 0.149i·13-s − 1.19i·14-s + (0.353 − 0.671i)15-s − 1.24·16-s + 0.453·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.785 + 0.618i$
Analytic conductor: \(11.8875\)
Root analytic conductor: \(3.44783\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :2),\ -0.785 + 0.618i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.787152 - 2.27050i\)
\(L(\frac12)\) \(\approx\) \(0.787152 - 2.27050i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-22.1 - 11.6i)T \)
23 \( 1 + (215. - 483. i)T \)
good2 \( 1 + 4.94iT - 16T^{2} \)
3 \( 1 + 6.82iT - 81T^{2} \)
7 \( 1 - 47.4T + 2.40e3T^{2} \)
11 \( 1 + 121. iT - 1.46e4T^{2} \)
13 \( 1 + 25.2iT - 2.85e4T^{2} \)
17 \( 1 - 130.T + 8.35e4T^{2} \)
19 \( 1 - 292. iT - 1.30e5T^{2} \)
29 \( 1 + 790.T + 7.07e5T^{2} \)
31 \( 1 - 365.T + 9.23e5T^{2} \)
37 \( 1 + 471.T + 1.87e6T^{2} \)
41 \( 1 + 1.82e3T + 2.82e6T^{2} \)
43 \( 1 + 7.12T + 3.41e6T^{2} \)
47 \( 1 + 2.54e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.85e3T + 7.89e6T^{2} \)
59 \( 1 + 4.05e3T + 1.21e7T^{2} \)
61 \( 1 - 3.18e3iT - 1.38e7T^{2} \)
67 \( 1 - 6.09e3T + 2.01e7T^{2} \)
71 \( 1 - 4.38e3T + 2.54e7T^{2} \)
73 \( 1 - 3.21e3iT - 2.83e7T^{2} \)
79 \( 1 - 8.82e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.28e4T + 4.74e7T^{2} \)
89 \( 1 - 5.49e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.55e4T + 8.85e7T^{2} \)
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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

−12.30062411078032024381886058081, −11.43095537603719224267943023859, −10.52678158913427873614714879902, −9.643705484868068059863746037935, −8.124130466876550462793722216709, −6.86977467356658360987381341671, −5.55387233804520840172023696281, −3.60158180686380508253842052346, −2.08394538685267049832454819759, −1.21154401734667787488633429571, 1.92115263247982645026398528751, 4.58188178672209420021986748322, 5.17419282107707371206942225397, 6.56847892459730308029660401646, 7.72437622903642626529742228213, 8.892116920432885198930052820999, 9.855694649400579874405921151474, 10.96783168266545128689222811100, 12.37670578513629432501002104182, 13.64972617176079384860924914165

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