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Properties

Label 2-605-55.49-c1-0-18
Degree $2$
Conductor $605$
Sign $0.996 + 0.0849i$
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  + (−1.94 + 2.68i)3-s + (−1.61 + 1.17i)4-s + (−1.11 − 1.93i)5-s + (−2.47 − 7.60i)9-s − 6.63i·12-s + (7.37 + 0.791i)15-s + (1.23 − 3.80i)16-s + (4.08 + 1.82i)20-s + 3.31i·23-s + (−2.51 + 4.31i)25-s + (15.7 + 5.12i)27-s + (1.54 + 4.75i)31-s + (12.9 + 9.40i)36-s + (−5.84 − 8.04i)37-s + (−12.0 + 13.2i)45-s + ⋯
L(s)  = 1  + (−1.12 + 1.54i)3-s + (−0.809 + 0.587i)4-s + (−0.498 − 0.867i)5-s + (−0.824 − 2.53i)9-s − 1.91i·12-s + (1.90 + 0.204i)15-s + (0.309 − 0.951i)16-s + (0.912 + 0.408i)20-s + 0.691i·23-s + (−0.503 + 0.863i)25-s + (3.03 + 0.986i)27-s + (0.277 + 0.854i)31-s + (2.15 + 1.56i)36-s + (−0.961 − 1.32i)37-s + (−1.78 + 1.97i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.996 + 0.0849i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.996 + 0.0849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.475450 - 0.0202212i\)
\(L(\frac12)\) \(\approx\) \(0.475450 - 0.0202212i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.11 + 1.93i)T \)
11 \( 1 \)
good2 \( 1 + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.94 - 2.68i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.31iT - 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.54 - 4.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.84 + 8.04i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-3.89 + 5.36i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-12.6 + 4.09i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-12.1 + 8.81i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 9.94iT - 67T^{2} \)
71 \( 1 + (0.927 - 2.85i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (9.46 - 3.07i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.56835527679734964704168483560, −9.747967539358031238133704881172, −9.017107427198352175951993059180, −8.400914554193036481189983874676, −7.02900848766867160582979054085, −5.47330794583931735854907906443, −5.11307324409018985408303382809, −4.06803189888003123442656227330, −3.57362900969374760079999479711, −0.42203143988934147925154091636, 0.967322388999460005605743076272, 2.47666909111751719761806772299, 4.28419428340696965942116992930, 5.43989763812236097768242718734, 6.21407248291865167890734844006, 6.95536906355256081472849092532, 7.81073701159315490018727534439, 8.669753937126066329134919883187, 10.18249999996844137378383895057, 10.73085383759179616391680843679

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