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Properties

Label 2-115-5.4-c3-0-10
Degree $2$
Conductor $115$
Sign $-0.932 - 0.360i$
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  + 1.45i·2-s + 7.24i·3-s + 5.88·4-s + (−4.02 + 10.4i)5-s − 10.5·6-s + 0.356i·7-s + 20.1i·8-s − 25.4·9-s + (−15.1 − 5.85i)10-s + 29.3·11-s + 42.6i·12-s − 41.0i·13-s − 0.518·14-s + (−75.5 − 29.1i)15-s + 17.7·16-s − 27.9i·17-s + ⋯
L(s)  = 1  + 0.513i·2-s + 1.39i·3-s + 0.736·4-s + (−0.360 + 0.932i)5-s − 0.715·6-s + 0.0192i·7-s + 0.891i·8-s − 0.942·9-s + (−0.479 − 0.185i)10-s + 0.804·11-s + 1.02i·12-s − 0.876i·13-s − 0.00989·14-s + (−1.30 − 0.501i)15-s + 0.277·16-s − 0.398i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.932 - 0.360i$
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: \((2,\ 115,\ (\ :3/2),\ -0.932 - 0.360i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.325017 + 1.74408i\)
\(L(\frac12)\) \(\approx\) \(0.325017 + 1.74408i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.02 - 10.4i)T \)
23 \( 1 - 23iT \)
good2 \( 1 - 1.45iT - 8T^{2} \)
3 \( 1 - 7.24iT - 27T^{2} \)
7 \( 1 - 0.356iT - 343T^{2} \)
11 \( 1 - 29.3T + 1.33e3T^{2} \)
13 \( 1 + 41.0iT - 2.19e3T^{2} \)
17 \( 1 + 27.9iT - 4.91e3T^{2} \)
19 \( 1 + 68.2T + 6.85e3T^{2} \)
29 \( 1 - 87.8T + 2.43e4T^{2} \)
31 \( 1 + 36.4T + 2.97e4T^{2} \)
37 \( 1 + 88.7iT - 5.06e4T^{2} \)
41 \( 1 - 502.T + 6.89e4T^{2} \)
43 \( 1 - 350. iT - 7.95e4T^{2} \)
47 \( 1 + 432. iT - 1.03e5T^{2} \)
53 \( 1 - 469. iT - 1.48e5T^{2} \)
59 \( 1 + 217.T + 2.05e5T^{2} \)
61 \( 1 + 167.T + 2.26e5T^{2} \)
67 \( 1 + 23.0iT - 3.00e5T^{2} \)
71 \( 1 - 824.T + 3.57e5T^{2} \)
73 \( 1 + 252. iT - 3.89e5T^{2} \)
79 \( 1 - 698.T + 4.93e5T^{2} \)
83 \( 1 - 1.06e3iT - 5.71e5T^{2} \)
89 \( 1 + 609.T + 7.04e5T^{2} \)
97 \( 1 + 649. iT - 9.12e5T^{2} \)
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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

−14.10526166719620634798960330540, −12.23189269767595302016284290837, −11.05891878404516740415396636441, −10.61508144048748343685463840046, −9.426703562445125733274386465089, −8.032775297342652423013834023295, −6.83816704326030751838946117122, −5.68612819285118978220912094274, −4.12544099624360017280907160804, −2.83408596535869628319276240949, 1.01294436944584412931479811018, 2.10652096030798137824927214167, 4.10037020671944852060217277442, 6.15821554538874251672161212671, 7.02487210654069730070428451629, 8.131612354172412699136844483384, 9.309545665174134108812822417248, 10.92403513490880454989930914811, 11.96975654183170318083807672010, 12.38905442675502208747338111942

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