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Properties

Label 2-115-5.3-c2-0-21
Degree $2$
Conductor $115$
Sign $-0.998 - 0.0569i$
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  + (1.71 − 1.71i)2-s + (−3.25 − 3.25i)3-s − 1.86i·4-s + (−4.79 − 1.42i)5-s − 11.1·6-s + (2.88 − 2.88i)7-s + (3.66 + 3.66i)8-s + 12.1i·9-s + (−10.6 + 5.76i)10-s − 12.0·11-s + (−6.05 + 6.05i)12-s + (−12.4 − 12.4i)13-s − 9.86i·14-s + (10.9 + 20.2i)15-s + 19.9·16-s + (21.2 − 21.2i)17-s + ⋯
L(s)  = 1  + (0.855 − 0.855i)2-s + (−1.08 − 1.08i)3-s − 0.465i·4-s + (−0.958 − 0.284i)5-s − 1.85·6-s + (0.411 − 0.411i)7-s + (0.457 + 0.457i)8-s + 1.35i·9-s + (−1.06 + 0.576i)10-s − 1.09·11-s + (−0.504 + 0.504i)12-s + (−0.961 − 0.961i)13-s − 0.704i·14-s + (0.730 + 1.34i)15-s + 1.24·16-s + (1.25 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0569i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.998 - 0.0569i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ -0.998 - 0.0569i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0306668 + 1.07564i\)
\(L(\frac12)\) \(\approx\) \(0.0306668 + 1.07564i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.79 + 1.42i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good2 \( 1 + (-1.71 + 1.71i)T - 4iT^{2} \)
3 \( 1 + (3.25 + 3.25i)T + 9iT^{2} \)
7 \( 1 + (-2.88 + 2.88i)T - 49iT^{2} \)
11 \( 1 + 12.0T + 121T^{2} \)
13 \( 1 + (12.4 + 12.4i)T + 169iT^{2} \)
17 \( 1 + (-21.2 + 21.2i)T - 289iT^{2} \)
19 \( 1 + 8.56iT - 361T^{2} \)
29 \( 1 + 41.0iT - 841T^{2} \)
31 \( 1 - 6.48T + 961T^{2} \)
37 \( 1 + (-14.3 + 14.3i)T - 1.36e3iT^{2} \)
41 \( 1 + 26.2T + 1.68e3T^{2} \)
43 \( 1 + (-41.3 - 41.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (12.4 - 12.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (4.51 + 4.51i)T + 2.80e3iT^{2} \)
59 \( 1 - 48.3iT - 3.48e3T^{2} \)
61 \( 1 - 89.1T + 3.72e3T^{2} \)
67 \( 1 + (-41.9 + 41.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 100.T + 5.04e3T^{2} \)
73 \( 1 + (-34.7 - 34.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 69.4iT - 6.24e3T^{2} \)
83 \( 1 + (81.6 + 81.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 53.2iT - 7.92e3T^{2} \)
97 \( 1 + (-106. + 106. i)T - 9.40e3iT^{2} \)
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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

−12.68895079640322055203874456891, −11.82913021406457126417058687983, −11.33594029292396477849441077133, −10.24144805800172497586006972620, −7.68221950168710697718364292320, −7.61345431735882963927360327378, −5.48859063212663921384592409759, −4.69794216978951377198003136403, −2.82324552942026980066946616655, −0.68738082524536044905615791999, 3.79994141451987811696053481097, 4.88614326756161286926567983146, 5.57519877885699709128609156776, 6.93251566997441627737618155896, 8.141413460673175631741166491100, 10.01786906239878593570674259642, 10.72845019238742096714010288514, 11.88351861789290505926340165833, 12.65798012061106800426282249355, 14.41891503163859693889837784876

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