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Properties

Label 2-605-11.3-c1-0-21
Degree $2$
Conductor $605$
Sign $0.927 - 0.374i$
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  + (−0.386 + 0.280i)2-s + (0.0998 + 0.307i)3-s + (−0.547 + 1.68i)4-s + (0.809 + 0.587i)5-s + (−0.124 − 0.0906i)6-s + (0.829 − 2.55i)7-s + (−0.556 − 1.71i)8-s + (2.34 − 1.70i)9-s − 0.477·10-s − 0.572·12-s + (3.77 − 2.74i)13-s + (0.396 + 1.21i)14-s + (−0.0998 + 0.307i)15-s + (−2.17 − 1.57i)16-s + (3.74 + 2.71i)17-s + (−0.427 + 1.31i)18-s + ⋯
L(s)  = 1  + (−0.273 + 0.198i)2-s + (0.0576 + 0.177i)3-s + (−0.273 + 0.842i)4-s + (0.361 + 0.262i)5-s + (−0.0509 − 0.0369i)6-s + (0.313 − 0.965i)7-s + (−0.196 − 0.605i)8-s + (0.780 − 0.567i)9-s − 0.150·10-s − 0.165·12-s + (1.04 − 0.760i)13-s + (0.105 + 0.325i)14-s + (−0.0257 + 0.0793i)15-s + (−0.543 − 0.394i)16-s + (0.907 + 0.659i)17-s + (−0.100 + 0.309i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43918 + 0.279540i\)
\(L(\frac12)\) \(\approx\) \(1.43918 + 0.279540i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.386 - 0.280i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.0998 - 0.307i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.829 + 2.55i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.77 + 2.74i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.74 - 2.71i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.34 + 4.12i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.77T + 23T^{2} \)
29 \( 1 + (0.931 - 2.86i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.93 - 1.40i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.28 - 10.1i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.683 + 2.10i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.06T + 43T^{2} \)
47 \( 1 + (-1.34 - 4.14i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.12 + 3.72i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.63 + 11.1i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.22 - 2.34i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 7.31T + 67T^{2} \)
71 \( 1 + (0.967 + 0.702i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.315 + 0.971i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.83 - 2.05i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.99 + 6.53i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 2.76T + 89T^{2} \)
97 \( 1 + (14.9 - 10.8i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.51767156487161817240315933434, −9.896872302633829465125913482551, −8.846714573877724117037379324448, −8.115051340636519733775168558866, −7.14918961837480264728341293084, −6.52686793568965118731490825361, −5.05202565302290802771944312582, −3.88591782207162999420776325736, −3.23538216431884904637273217050, −1.14726560832438018287757879647, 1.36353183273267094896567968572, 2.23501505122316450095146585375, 4.08106476554861232942774219467, 5.29502888748963709042368709243, 5.80953002511657124400524855824, 6.99687072024679066846846758678, 8.246496482343653326099874774538, 8.936241718284262970541982795255, 9.716142906416616443531655152202, 10.47754661342816907030980772503

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