↑

Properties

Label 2-115-23.2-c1-0-2
Degree $2$
Conductor $115$
Sign $0.319 - 0.947i$
Analytic cond. $0.918279$
Root an. cond. $0.958269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.698 + 0.449i)2-s + (0.399 + 2.78i)3-s + (−0.544 − 1.19i)4-s + (0.959 + 0.281i)5-s + (−0.969 + 2.12i)6-s + (−0.131 + 0.151i)7-s + (0.391 − 2.72i)8-s + (−4.70 + 1.38i)9-s + (0.544 + 0.627i)10-s + (−1.37 + 0.881i)11-s + (3.09 − 1.99i)12-s + (0.350 + 0.404i)13-s + (−0.159 + 0.0468i)14-s + (−0.399 + 2.78i)15-s + (−0.219 + 0.253i)16-s + (3.00 − 6.57i)17-s + ⋯
L(s)  = 1  + (0.494 + 0.317i)2-s + (0.230 + 1.60i)3-s + (−0.272 − 0.595i)4-s + (0.429 + 0.125i)5-s + (−0.395 + 0.867i)6-s + (−0.0495 + 0.0571i)7-s + (0.138 − 0.962i)8-s + (−1.56 + 0.460i)9-s + (0.172 + 0.198i)10-s + (−0.413 + 0.265i)11-s + (0.893 − 0.574i)12-s + (0.0972 + 0.112i)13-s + (−0.0426 + 0.0125i)14-s + (−0.103 + 0.718i)15-s + (−0.0548 + 0.0632i)16-s + (0.728 − 1.59i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.319 - 0.947i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ 0.319 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09570 + 0.787236i\)
\(L(\frac12)\) \(\approx\) \(1.09570 + 0.787236i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (-4.19 - 2.33i)T \)
good2 \( 1 + (-0.698 - 0.449i)T + (0.830 + 1.81i)T^{2} \)
3 \( 1 + (-0.399 - 2.78i)T + (-2.87 + 0.845i)T^{2} \)
7 \( 1 + (0.131 - 0.151i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (1.37 - 0.881i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-0.350 - 0.404i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-3.00 + 6.57i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (1.79 + 3.93i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (-1.23 + 2.69i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.651 - 4.53i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (5.70 - 1.67i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (11.7 + 3.45i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-1.00 - 7.01i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + (-4.66 + 5.38i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-4.21 - 4.86i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (-0.499 + 3.47i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (-13.4 - 8.65i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (4.49 + 2.88i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-1.57 - 3.44i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (7.73 + 8.92i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-5.40 + 1.58i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (0.166 + 1.15i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (-4.83 - 1.41i)T + (81.6 + 52.4i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.05990935585245481349358257520, −13.16618578403769692545983793740, −11.45150970051206144253543081494, −10.31107739359351183715080044151, −9.712494540059842138035822402481, −8.850109147177963679395320430393, −6.88850128790767051941362993575, −5.29421066344392772717545797716, −4.78222269595058174974456389092, −3.20999085756290931290778988023, 1.92057895864577741253382987939, 3.45647756125161459986461741455, 5.46027717893694481802537503652, 6.71468309889591242957672201517, 8.044008975898090630030197998273, 8.555080634006846445198702753304, 10.43491952275766179117921888661, 11.81314608391413645750242388940, 12.70369774063782925142204022885, 13.07158353671320179863939711519

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