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Properties

Label 2-115-5.4-c3-0-1
Degree $2$
Conductor $115$
Sign $0.167 + 0.985i$
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  + 5.07i·2-s + 3.81i·3-s − 17.7·4-s + (−11.0 + 1.87i)5-s − 19.3·6-s + 14.6i·7-s − 49.3i·8-s + 12.4·9-s + (−9.49 − 55.9i)10-s + 27.8·11-s − 67.6i·12-s − 3.74i·13-s − 74.5·14-s + (−7.15 − 42.0i)15-s + 108.·16-s + 6.69i·17-s + ⋯
L(s)  = 1  + 1.79i·2-s + 0.734i·3-s − 2.21·4-s + (−0.985 + 0.167i)5-s − 1.31·6-s + 0.793i·7-s − 2.18i·8-s + 0.459·9-s + (−0.300 − 1.76i)10-s + 0.762·11-s − 1.62i·12-s − 0.0799i·13-s − 1.42·14-s + (−0.123 − 0.724i)15-s + 1.69·16-s + 0.0954i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.167 + 0.985i$
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: \((2,\ 115,\ (\ :3/2),\ 0.167 + 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.597247 - 0.504336i\)
\(L(\frac12)\) \(\approx\) \(0.597247 - 0.504336i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (11.0 - 1.87i)T \)
23 \( 1 + 23iT \)
good2 \( 1 - 5.07iT - 8T^{2} \)
3 \( 1 - 3.81iT - 27T^{2} \)
7 \( 1 - 14.6iT - 343T^{2} \)
11 \( 1 - 27.8T + 1.33e3T^{2} \)
13 \( 1 + 3.74iT - 2.19e3T^{2} \)
17 \( 1 - 6.69iT - 4.91e3T^{2} \)
19 \( 1 + 132.T + 6.85e3T^{2} \)
29 \( 1 + 76.1T + 2.43e4T^{2} \)
31 \( 1 + 50.4T + 2.97e4T^{2} \)
37 \( 1 - 288. iT - 5.06e4T^{2} \)
41 \( 1 + 254.T + 6.89e4T^{2} \)
43 \( 1 + 209. iT - 7.95e4T^{2} \)
47 \( 1 + 63.5iT - 1.03e5T^{2} \)
53 \( 1 - 624. iT - 1.48e5T^{2} \)
59 \( 1 + 796.T + 2.05e5T^{2} \)
61 \( 1 - 676.T + 2.26e5T^{2} \)
67 \( 1 - 933. iT - 3.00e5T^{2} \)
71 \( 1 + 791.T + 3.57e5T^{2} \)
73 \( 1 + 5.71iT - 3.89e5T^{2} \)
79 \( 1 - 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 1.03e3iT - 5.71e5T^{2} \)
89 \( 1 - 649.T + 7.04e5T^{2} \)
97 \( 1 + 308. iT - 9.12e5T^{2} \)
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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

−14.56868276976681958563978117542, −13.12467197042407701043251210979, −11.98925325680743582947851586750, −10.49375444001460786879139809043, −9.112350429069686064631545273294, −8.451577692520540129973875815042, −7.22770254929498640957379238175, −6.22460281962123912195370176777, −4.81025400328535903098052470668, −3.90087897760446063035371755092, 0.44267605163385127086531830407, 1.78351564596704610444472125356, 3.67609888653742211005445669601, 4.46311113732176892749361313426, 6.86624671236910312476765472075, 8.123035607799410241504858490271, 9.307694029953013942329179003710, 10.55278187812209404103592388546, 11.31836545729696522594876519561, 12.30222931666239110648422976806

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