Properties

Label 2-115-23.22-c2-0-5
Degree $2$
Conductor $115$
Sign $0.998 + 0.0586i$
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-s − 4.73·3-s − 3·4-s + 2.23i·5-s + 4.73·6-s − 7.39i·7-s + 7·8-s + 13.3·9-s − 2.23i·10-s + 16.9i·11-s + 14.1·12-s + 13.8·13-s + 7.39i·14-s − 10.5i·15-s + 5·16-s − 30.3i·17-s + ⋯
L(s)  = 1  − 0.5·2-s − 1.57·3-s − 0.750·4-s + 0.447i·5-s + 0.788·6-s − 1.05i·7-s + 0.875·8-s + 1.48·9-s − 0.223i·10-s + 1.54i·11-s + 1.18·12-s + 1.06·13-s + 0.528i·14-s − 0.705i·15-s + 0.312·16-s − 1.78i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0586i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.998 + 0.0586i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ 0.998 + 0.0586i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.512635 - 0.0150394i\)
\(L(\frac12)\) \(\approx\) \(0.512635 - 0.0150394i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 2.23iT \)
23 \( 1 + (1.34 - 22.9i)T \)
good2 \( 1 + T + 4T^{2} \)
3 \( 1 + 4.73T + 9T^{2} \)
7 \( 1 + 7.39iT - 49T^{2} \)
11 \( 1 - 16.9iT - 121T^{2} \)
13 \( 1 - 13.8T + 169T^{2} \)
17 \( 1 + 30.3iT - 289T^{2} \)
19 \( 1 + 6.01iT - 361T^{2} \)
29 \( 1 - 42.3T + 841T^{2} \)
31 \( 1 + 17.1T + 961T^{2} \)
37 \( 1 - 28.2iT - 1.36e3T^{2} \)
41 \( 1 - 69.8T + 1.68e3T^{2} \)
43 \( 1 + 17.7iT - 1.84e3T^{2} \)
47 \( 1 - 49.6T + 2.20e3T^{2} \)
53 \( 1 + 40.2iT - 2.80e3T^{2} \)
59 \( 1 - 40.6T + 3.48e3T^{2} \)
61 \( 1 + 2.84iT - 3.72e3T^{2} \)
67 \( 1 + 35.2iT - 4.48e3T^{2} \)
71 \( 1 - 9.88T + 5.04e3T^{2} \)
73 \( 1 - 30.3T + 5.32e3T^{2} \)
79 \( 1 + 98.0iT - 6.24e3T^{2} \)
83 \( 1 - 100. iT - 6.88e3T^{2} \)
89 \( 1 - 28.3iT - 7.92e3T^{2} \)
97 \( 1 - 174. iT - 9.40e3T^{2} \)
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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

−13.28217640392084342784484652615, −12.08392048581502867664903142295, −11.04760419777743414100453770671, −10.26730079135451891606097799817, −9.422820288852821120509111946251, −7.54407357553127710414174617680, −6.78277686489362553250208225272, −5.18590035639529922685407344005, −4.21992718113003993807929128462, −0.858239875112471960930449468348, 0.908929676917361942850124067146, 4.16391628642549734850806123138, 5.66882003287649888990913127039, 6.08670852181598725326393209135, 8.306423487062263852108493501724, 8.872431370260496519319599986712, 10.45584256590916977536916975264, 11.06873497994496590086041120915, 12.31714132582330730839261860444, 12.95834042881053285720713009037

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