Properties

Label 2-605-55.14-c1-0-42
Degree $2$
Conductor $605$
Sign $0.396 - 0.917i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (−1.48 − 2.04i)2-s + (0.753 − 0.244i)3-s + (−1.35 + 4.15i)4-s + (0.695 − 2.12i)5-s + (−1.61 − 1.17i)6-s + (−3.29 − 1.07i)7-s + (5.69 − 1.85i)8-s + (−1.91 + 1.39i)9-s + (−5.37 + 1.73i)10-s + 3.46i·12-s + (2.70 + 8.31i)14-s + (0.00403 − 1.77i)15-s + (−5.15 − 3.74i)16-s + (−0.931 + 1.28i)17-s + (5.69 + 1.85i)18-s + (−1.23 − 3.80i)19-s + ⋯
L(s)  = 1  + (−1.04 − 1.44i)2-s + (0.435 − 0.141i)3-s + (−0.675 + 2.07i)4-s + (0.311 − 0.950i)5-s + (−0.660 − 0.479i)6-s + (−1.24 − 0.404i)7-s + (2.01 − 0.654i)8-s + (−0.639 + 0.464i)9-s + (−1.69 + 0.547i)10-s + 0.999i·12-s + (0.722 + 2.22i)14-s + (0.00104 − 0.457i)15-s + (−1.28 − 0.936i)16-s + (−0.225 + 0.310i)17-s + (1.34 + 0.436i)18-s + (−0.283 − 0.872i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.396 - 0.917i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.396 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0400128 + 0.0262964i\)
\(L(\frac12)\) \(\approx\) \(0.0400128 + 0.0262964i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.695 + 2.12i)T \)
11 \( 1 \)
good2 \( 1 + (1.48 + 2.04i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.753 + 0.244i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (3.29 + 1.07i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.931 - 1.28i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.23 + 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.792iT - 23T^{2} \)
29 \( 1 + (2.70 - 8.31i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.72 - 1.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.03 - 0.335i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.70 - 8.31i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + (6.30 - 2.04i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.93 + 8.16i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.27 + 7.01i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.602 - 0.437i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.30iT - 67T^{2} \)
71 \( 1 + (-8.18 - 5.94i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.58 + 2.14i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.01 + 0.737i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.89 + 5.36i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 + (-3.43 - 4.72i)T + (-29.9 + 92.2i)T^{2} \)
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

−9.747573227277168553893319056373, −9.278054664700762478445894887552, −8.576405755377622283678238661341, −7.79657589273989808935676945434, −6.51772867953774652340693243846, −5.00658564339234245357515815858, −3.64874525639061503518473993899, −2.78728646146641804109768036256, −1.57711174081518398953252397732, −0.03431532570515178041576035530, 2.47989192780836661283370721821, 3.74859620379719581122547432828, 5.73074838803678499985284379347, 6.10968162504045269054238298613, 6.93234144242676395613846799682, 7.81043835902885048694921691217, 8.758929373600431431080367242051, 9.523555193526367631797077775193, 9.910117828019619229515410145255, 10.91853070333500600379201567565

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