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Properties

Label 2-115-5.4-c3-0-5
Degree $2$
Conductor $115$
Sign $0.996 - 0.0840i$
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  − 5.19i·2-s + 7.60i·3-s − 18.9·4-s + (−0.939 − 11.1i)5-s + 39.4·6-s + 25.9i·7-s + 56.7i·8-s − 30.8·9-s + (−57.8 + 4.87i)10-s + 46.1·11-s − 144. i·12-s + 67.2i·13-s + 134.·14-s + (84.7 − 7.15i)15-s + 143.·16-s + 20.2i·17-s + ⋯
L(s)  = 1  − 1.83i·2-s + 1.46i·3-s − 2.36·4-s + (−0.0840 − 0.996i)5-s + 2.68·6-s + 1.40i·7-s + 2.50i·8-s − 1.14·9-s + (−1.82 + 0.154i)10-s + 1.26·11-s − 3.46i·12-s + 1.43i·13-s + 2.57·14-s + (1.45 − 0.123i)15-s + 2.23·16-s + 0.288i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0840i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.996 - 0.0840i$
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: \((2,\ 115,\ (\ :3/2),\ 0.996 - 0.0840i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.20948 + 0.0509334i\)
\(L(\frac12)\) \(\approx\) \(1.20948 + 0.0509334i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.939 + 11.1i)T \)
23 \( 1 - 23iT \)
good2 \( 1 + 5.19iT - 8T^{2} \)
3 \( 1 - 7.60iT - 27T^{2} \)
7 \( 1 - 25.9iT - 343T^{2} \)
11 \( 1 - 46.1T + 1.33e3T^{2} \)
13 \( 1 - 67.2iT - 2.19e3T^{2} \)
17 \( 1 - 20.2iT - 4.91e3T^{2} \)
19 \( 1 - 87.1T + 6.85e3T^{2} \)
29 \( 1 + 28.5T + 2.43e4T^{2} \)
31 \( 1 + 320.T + 2.97e4T^{2} \)
37 \( 1 - 226. iT - 5.06e4T^{2} \)
41 \( 1 - 255.T + 6.89e4T^{2} \)
43 \( 1 - 9.43iT - 7.95e4T^{2} \)
47 \( 1 + 220. iT - 1.03e5T^{2} \)
53 \( 1 - 56.8iT - 1.48e5T^{2} \)
59 \( 1 + 82.0T + 2.05e5T^{2} \)
61 \( 1 - 236.T + 2.26e5T^{2} \)
67 \( 1 - 163. iT - 3.00e5T^{2} \)
71 \( 1 - 114.T + 3.57e5T^{2} \)
73 \( 1 + 109. iT - 3.89e5T^{2} \)
79 \( 1 - 1.26e3T + 4.93e5T^{2} \)
83 \( 1 + 921. iT - 5.71e5T^{2} \)
89 \( 1 + 813.T + 7.04e5T^{2} \)
97 \( 1 + 1.49e3iT - 9.12e5T^{2} \)
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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

−12.59694466549123750532299793343, −11.75814015078780846871765214724, −11.30200264919315384928012416697, −9.755298836035652288040972031792, −9.179177610118332587254489761588, −8.825143396453747955237874673842, −5.48085190628051324682199339790, −4.45584406355060223364835132518, −3.56132663625361186768737826526, −1.74275167402282923323272099300, 0.71614864863924502859499272701, 3.74523683823342995146719672652, 5.72612835848532476626473200033, 6.77279584081172139081165550328, 7.35331685043574930144765983901, 7.890837820621492569735145450682, 9.501048511660515716437193710390, 10.97209725703642521868386989988, 12.57299389626163945113866606057, 13.55494871015562917504699676669

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