Properties

Label 2-605-11.4-c1-0-27
Degree $2$
Conductor $605$
Sign $-0.220 + 0.975i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.220 + 0.975i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.931238 - 1.16563i\)
\(L(\frac12)\) \(\approx\) \(0.931238 - 1.16563i\)
\(L(\frac{3}{2})\) 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

−10.54079579936502957563372391902, −9.625368356623776512442785459447, −8.829208881534159636717103430909, −7.33427545132113174904050187609, −6.99764419915312568209000029635, −5.70939649103102737806783302371, −4.85611757125981311747921320760, −4.03194684026880505042893605531, −2.24111826040546594877865454939, −0.76209708290372485518645155124, 2.16842133543638188016739592337, 3.14060590749210100741503158183, 4.30233020353358873128462496001, 5.17996043175499811229767837210, 6.51509367499774689370231775496, 7.24771873795029206432466608110, 8.397229427098756290824188034690, 9.375318822097699300154506404739, 9.702474786511365425277021699477, 11.05300600798835448790079750558

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