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Properties

Label 2-605-55.9-c1-0-43
Degree $2$
Conductor $605$
Sign $-0.605 + 0.795i$
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  + (1.26 + 0.410i)2-s + (−0.190 − 0.138i)4-s + (−0.690 − 2.12i)5-s + (−2.52 + 3.47i)7-s + (−1.74 − 2.40i)8-s + (0.927 − 2.85i)9-s − 2.96i·10-s + (−6.61 − 2.14i)13-s + (−4.61 + 3.35i)14-s + (−1.07 − 3.30i)16-s + (1.56 − 0.507i)17-s + (2.34 − 3.22i)18-s + (−0.163 + 0.502i)20-s + (−4.04 + 2.93i)25-s + (−7.47 − 5.42i)26-s + ⋯
L(s)  = 1  + (0.893 + 0.290i)2-s + (−0.0954 − 0.0693i)4-s + (−0.309 − 0.951i)5-s + (−0.954 + 1.31i)7-s + (−0.617 − 0.849i)8-s + (0.309 − 0.951i)9-s − 0.939i·10-s + (−1.83 − 0.596i)13-s + (−1.23 + 0.896i)14-s + (−0.268 − 0.825i)16-s + (0.378 − 0.123i)17-s + (0.552 − 0.759i)18-s + (−0.0364 + 0.112i)20-s + (−0.809 + 0.587i)25-s + (−1.46 − 1.06i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410450 - 0.827929i\)
\(L(\frac12)\) \(\approx\) \(0.410450 - 0.827929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.690 + 2.12i)T \)
11 \( 1 \)
good2 \( 1 + (-1.26 - 0.410i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (2.52 - 3.47i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (6.61 + 2.14i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.56 + 0.507i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.76 + 8.50i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.01iT - 43T^{2} \)
47 \( 1 + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.23 - 2.35i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (-2.47 - 7.60i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.21 + 9.92i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-17.3 + 5.62i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + (78.4 + 57.0i)T^{2} \)
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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

−9.803559373486114914522768094302, −9.629776701882979438410924852931, −8.783038387679251426161320853325, −7.54790486026267907886470605350, −6.40329049653772795924919794839, −5.60839469254896660641091009393, −4.90151566236108858789018939762, −3.81641808073273028376633041448, −2.69139613564784564134322976748, −0.36568054140630699982563828319, 2.43251737404647296561262193985, 3.39088455990360914138046953108, 4.29822570695035416665600697241, 5.12604591053759199882617032502, 6.60574808302489425402127998622, 7.24476737795086625190023568963, 8.041095906344173632000458496976, 9.563058010402889835372440048067, 10.24638357842833646696490630890, 10.94250668188230739477437616168

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