Properties

Label 2-605-11.9-c1-0-25
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: $\chi_{605} (251, \cdot )$
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

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

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