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Properties

Label 2-605-5.4-c1-0-43
Degree $2$
Conductor $605$
Sign $-0.977 - 0.210i$
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  − 0.792i·2-s − 2.52i·3-s + 1.37·4-s + (−2.18 − 0.469i)5-s − 2·6-s − 3.46i·7-s − 2.67i·8-s − 3.37·9-s + (−0.372 + 1.73i)10-s − 3.46i·12-s − 2.74·14-s + (−1.18 + 5.51i)15-s + 0.627·16-s + 5.04i·17-s + 2.67i·18-s + 4·19-s + ⋯
L(s)  = 1  − 0.560i·2-s − 1.45i·3-s + 0.686·4-s + (−0.977 − 0.210i)5-s − 0.816·6-s − 1.30i·7-s − 0.944i·8-s − 1.12·9-s + (−0.117 + 0.547i)10-s − 0.999i·12-s − 0.733·14-s + (−0.306 + 1.42i)15-s + 0.156·16-s + 1.22i·17-s + 0.629i·18-s + 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.977 - 0.210i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.977 - 0.210i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146639 + 1.38007i\)
\(L(\frac12)\) \(\approx\) \(0.146639 + 1.38007i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.18 + 0.469i)T \)
11 \( 1 \)
good2 \( 1 + 0.792iT - 2T^{2} \)
3 \( 1 + 2.52iT - 3T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5.04iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 2.52iT - 23T^{2} \)
29 \( 1 + 2.74T + 29T^{2} \)
31 \( 1 + 2.37T + 31T^{2} \)
37 \( 1 + 11.0iT - 37T^{2} \)
41 \( 1 - 2.74T + 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 - 6.63iT - 47T^{2} \)
53 \( 1 - 3.16iT - 53T^{2} \)
59 \( 1 - 1.62T + 59T^{2} \)
61 \( 1 + 10.7T + 61T^{2} \)
67 \( 1 + 0.644iT - 67T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + 6.63iT - 83T^{2} \)
89 \( 1 - 4.37T + 89T^{2} \)
97 \( 1 + 4.10iT - 97T^{2} \)
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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

−10.66677015585856138930471372173, −9.355445806207278041584819802711, −7.88569327122399121617862012997, −7.56167110682935819631897649123, −6.91189455124613568041877591161, −5.91036376415282172734571149765, −4.14738754503175398895066481644, −3.28458531625869575087816792307, −1.75634307146198927034850509194, −0.78589337587497959640231614419, 2.62445275935766940188297970435, 3.47675716629220954232490730031, 4.86680725215035031316690321782, 5.42657400093018570778408055405, 6.62839621550305437245631857300, 7.62312346452348710948298320952, 8.552125052468707136746348602842, 9.295072242858146378478686328783, 10.24765190727687604585737753710, 11.23463096596963189001057565436

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