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Properties

Label 2-115-115.114-c2-0-8
Degree $2$
Conductor $115$
Sign $0.709 - 0.704i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.709 - 0.704i$
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: \((2,\ 115,\ (\ :1),\ 0.709 - 0.704i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.62567 + 0.670489i\)
\(L(\frac12)\) \(\approx\) \(1.62567 + 0.670489i\)
\(L(2)\) 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

−13.49168647448242147272477795743, −12.04381675196505840086534618240, −11.31795380872785328238834822059, −10.26084608203551089605156498452, −9.439964232726483507752024341647, −8.373810201457263638067592688541, −6.65440583183823411928855952049, −4.94576119110899546518819205409, −4.31679443599479046413452700334, −2.19616038723195940833784955511, 1.66562852274511754432505332049, 2.92511016123566630142445542599, 5.93963029384119034245601682939, 6.42074907919530906778874214435, 7.71358519794924057571479046932, 8.180445845263698989553436702958, 10.41542880141630730973257389355, 11.35124961113313250291878338155, 12.00915225385665852762905545559, 13.34982097828282130748907586788

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