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Properties

Label 2-115-115.114-c2-0-21
Degree $2$
Conductor $115$
Sign $-0.700 - 0.713i$
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  − 3.64i·2-s − 3.06i·3-s − 9.32·4-s + (−3.24 − 3.80i)5-s − 11.1·6-s + 12.1·7-s + 19.4i·8-s − 0.397·9-s + (−13.8 + 11.8i)10-s + 10.9i·11-s + 28.5i·12-s − 17.9i·13-s − 44.4i·14-s + (−11.6 + 9.94i)15-s + 33.5·16-s + 0.374·17-s + ⋯
L(s)  = 1  − 1.82i·2-s − 1.02i·3-s − 2.33·4-s + (−0.648 − 0.761i)5-s − 1.86·6-s + 1.74·7-s + 2.42i·8-s − 0.0441·9-s + (−1.38 + 1.18i)10-s + 0.991i·11-s + 2.38i·12-s − 1.38i·13-s − 3.17i·14-s + (−0.777 + 0.662i)15-s + 2.09·16-s + 0.0220·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.700 - 0.713i$
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: \((2,\ 115,\ (\ :1),\ -0.700 - 0.713i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.468699 + 1.11704i\)
\(L(\frac12)\) \(\approx\) \(0.468699 + 1.11704i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (3.24 + 3.80i)T \)
23 \( 1 + (22.9 - 1.61i)T \)
good2 \( 1 + 3.64iT - 4T^{2} \)
3 \( 1 + 3.06iT - 9T^{2} \)
7 \( 1 - 12.1T + 49T^{2} \)
11 \( 1 - 10.9iT - 121T^{2} \)
13 \( 1 + 17.9iT - 169T^{2} \)
17 \( 1 - 0.374T + 289T^{2} \)
19 \( 1 - 10.3iT - 361T^{2} \)
29 \( 1 + 11.7T + 841T^{2} \)
31 \( 1 - 32.5T + 961T^{2} \)
37 \( 1 - 25.4T + 1.36e3T^{2} \)
41 \( 1 - 13.8T + 1.68e3T^{2} \)
43 \( 1 - 14.5T + 1.84e3T^{2} \)
47 \( 1 + 41.7iT - 2.20e3T^{2} \)
53 \( 1 + 68.2T + 2.80e3T^{2} \)
59 \( 1 - 9.15T + 3.48e3T^{2} \)
61 \( 1 + 63.7iT - 3.72e3T^{2} \)
67 \( 1 - 4.06T + 4.48e3T^{2} \)
71 \( 1 - 47.2T + 5.04e3T^{2} \)
73 \( 1 - 80.6iT - 5.32e3T^{2} \)
79 \( 1 - 74.1iT - 6.24e3T^{2} \)
83 \( 1 - 151.T + 6.88e3T^{2} \)
89 \( 1 - 84.2iT - 7.92e3T^{2} \)
97 \( 1 + 36.3T + 9.40e3T^{2} \)
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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

−12.38381059952230942161887529377, −11.90435324509770649807432255703, −10.91193203353578963752854458831, −9.779495023502815549818291979530, −8.193842318786790445451261815194, −7.83772644264706290370714762233, −5.14291357725308059407734591560, −4.13563088249261501683133190071, −2.06832611714376703597463255032, −1.01212044658234672960744398274, 4.11196896369744438437763254235, 4.77089915012508716891301211412, 6.21642196112793526780324781657, 7.49300226927770110844666207434, 8.303847401318511473958667798940, 9.318489780286810351063452360304, 10.78676900651423026230977140301, 11.65363626133302451517552118037, 13.78645405789150171160086453950, 14.39043163839262081708887725110

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