Properties

Label 2-605-11.5-c1-0-1
Degree $2$
Conductor $605$
Sign $0.263 - 0.964i$
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  + (−0.418 − 1.28i)2-s + (−0.465 − 0.338i)3-s + (0.131 − 0.0953i)4-s + (−0.309 + 0.951i)5-s + (−0.241 + 0.741i)6-s + (−2.95 + 2.14i)7-s + (−2.37 − 1.72i)8-s + (−0.824 − 2.53i)9-s + 1.35·10-s − 0.0933·12-s + (0.874 + 2.69i)13-s + (4.00 + 2.90i)14-s + (0.465 − 0.338i)15-s + (−1.12 + 3.47i)16-s + (−1.14 + 3.51i)17-s + (−2.92 + 2.12i)18-s + ⋯
L(s)  = 1  + (−0.296 − 0.911i)2-s + (−0.268 − 0.195i)3-s + (0.0655 − 0.0476i)4-s + (−0.138 + 0.425i)5-s + (−0.0984 + 0.302i)6-s + (−1.11 + 0.810i)7-s + (−0.838 − 0.609i)8-s + (−0.274 − 0.846i)9-s + 0.428·10-s − 0.0269·12-s + (0.242 + 0.746i)13-s + (1.06 + 0.777i)14-s + (0.120 − 0.0873i)15-s + (−0.281 + 0.867i)16-s + (−0.276 + 0.851i)17-s + (−0.689 + 0.501i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.263 - 0.964i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.263 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.244714 + 0.186842i\)
\(L(\frac12)\) \(\approx\) \(0.244714 + 0.186842i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (0.418 + 1.28i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.465 + 0.338i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (2.95 - 2.14i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.874 - 2.69i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.14 - 3.51i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.0769 + 0.0559i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 + (5.46 - 3.96i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.09 - 6.44i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.96 - 5.78i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.72 + 4.88i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.96T + 43T^{2} \)
47 \( 1 + (-1.79 - 1.30i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.925 - 2.84i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.88 + 5.00i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.62 + 8.06i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + (-2.56 + 7.89i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.06 - 0.775i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.28 - 13.1i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.28 + 10.1i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (1.33 + 4.11i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.86335561627544301458660749941, −10.08483901810932481625504072780, −9.182821583633509457498070929211, −8.699818262700436626421100411724, −6.84688255174900889381598707906, −6.51521294034775634799761656435, −5.57224411186557502739796976858, −3.68644025371325613403267508495, −3.02290619402642691214279821999, −1.69736431063964018481726433243, 0.18195150161466740869011237138, 2.63812723761394884307158217394, 3.88307999277936950770045276328, 5.23062667042090140249421419411, 5.99073242904696390292938082022, 7.03902264317442229100661232895, 7.65900162315775146964799784256, 8.553279824332233970262953708370, 9.500653728825121104335006172045, 10.37268523434410534051835139786

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