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Properties

Label 2-605-55.4-c1-0-0
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

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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: $\chi_{605} (444, \cdot )$
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 \( 1 + (-3.43 + 4.72i)T + (-29.9 - 92.2i)T^{2} \)
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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

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

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