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Properties

Label 2-115-115.114-c2-0-6
Degree $2$
Conductor $115$
Sign $-0.784 - 0.619i$
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  + 2.13i·2-s + 3.84i·3-s − 0.566·4-s + (4.79 + 1.42i)5-s − 8.21·6-s + 2.33·7-s + 7.33i·8-s − 5.77·9-s + (−3.05 + 10.2i)10-s − 12.3i·11-s − 2.17i·12-s − 19.6i·13-s + 4.99i·14-s + (−5.49 + 18.4i)15-s − 17.9·16-s − 24.3·17-s + ⋯
L(s)  = 1  + 1.06i·2-s + 1.28i·3-s − 0.141·4-s + (0.958 + 0.285i)5-s − 1.36·6-s + 0.334·7-s + 0.917i·8-s − 0.641·9-s + (−0.305 + 1.02i)10-s − 1.12i·11-s − 0.181i·12-s − 1.51i·13-s + 0.356i·14-s + (−0.366 + 1.22i)15-s − 1.12·16-s − 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.619i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.784 - 0.619i$
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: \((2,\ 115,\ (\ :1),\ -0.784 - 0.619i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.560341 + 1.61304i\)
\(L(\frac12)\) \(\approx\) \(0.560341 + 1.61304i\)
\(L(2)\) 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 \( 1 + (-21.3 - 8.50i)T \)
good2 \( 1 - 2.13iT - 4T^{2} \)
3 \( 1 - 3.84iT - 9T^{2} \)
7 \( 1 - 2.33T + 49T^{2} \)
11 \( 1 + 12.3iT - 121T^{2} \)
13 \( 1 + 19.6iT - 169T^{2} \)
17 \( 1 + 24.3T + 289T^{2} \)
19 \( 1 + 19.9iT - 361T^{2} \)
29 \( 1 - 8.59T + 841T^{2} \)
31 \( 1 + 2.00T + 961T^{2} \)
37 \( 1 + 61.1T + 1.36e3T^{2} \)
41 \( 1 - 27.4T + 1.68e3T^{2} \)
43 \( 1 - 37.3T + 1.84e3T^{2} \)
47 \( 1 - 16.3iT - 2.20e3T^{2} \)
53 \( 1 + 44.5T + 2.80e3T^{2} \)
59 \( 1 - 58.7T + 3.48e3T^{2} \)
61 \( 1 - 39.7iT - 3.72e3T^{2} \)
67 \( 1 + 44.9T + 4.48e3T^{2} \)
71 \( 1 - 15.8T + 5.04e3T^{2} \)
73 \( 1 + 101. iT - 5.32e3T^{2} \)
79 \( 1 - 138. iT - 6.24e3T^{2} \)
83 \( 1 - 30.6T + 6.88e3T^{2} \)
89 \( 1 - 53.2iT - 7.92e3T^{2} \)
97 \( 1 + 168.T + 9.40e3T^{2} \)
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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

−14.04989254931339731324276744097, −13.15010548553395597347132995018, −10.99481145986088476212460851123, −10.78922629146437918337862721886, −9.299733960875577864200299906503, −8.460658990920279589708130297271, −6.93044707994036519468557443886, −5.71242892450028195111170534592, −4.92144983953944956123079623239, −2.92178732943725507080453658673, 1.59705474344725069677778432855, 2.19322097007038613228231314524, 4.50461590425137537173992903877, 6.46827211530279732218537823183, 7.11052245408838103985767549302, 8.841905571461074282873832199476, 9.864838369812822325090781738748, 11.03956162998472192284905633559, 12.11652698898620159355354753953, 12.70794011151661399430116011377

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