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Properties

Label 2-605-11.5-c1-0-5
Degree $2$
Conductor $605$
Sign $-0.605 - 0.795i$
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  + (0.535 + 1.64i)2-s + (−1.61 − 1.17i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (1.07 − 3.29i)6-s + (−2.80 + 2.03i)7-s + (1.40 + 1.01i)8-s + (0.309 + 0.951i)9-s + 1.73·10-s + 1.99·12-s + (−4.85 − 3.52i)14-s + (−1.61 + 1.17i)15-s + (−1.54 + 4.75i)16-s + (−2.14 + 6.58i)17-s + (−1.40 + 1.01i)18-s + (5.60 + 4.07i)19-s + ⋯
L(s)  = 1  + (0.378 + 1.16i)2-s + (−0.934 − 0.678i)3-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (0.437 − 1.34i)6-s + (−1.05 + 0.769i)7-s + (0.495 + 0.359i)8-s + (0.103 + 0.317i)9-s + 0.547·10-s + 0.577·12-s + (−1.29 − 0.942i)14-s + (−0.417 + 0.303i)15-s + (−0.386 + 1.18i)16-s + (−0.519 + 1.59i)17-s + (−0.330 + 0.239i)18-s + (1.28 + 0.934i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.605 - 0.795i$
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: \((2,\ 605,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.470709 + 0.949479i\)
\(L(\frac12)\) \(\approx\) \(0.470709 + 0.949479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.535 - 1.64i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.61 + 1.17i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (2.80 - 2.03i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.14 - 6.58i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.60 - 4.07i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.23 - 3.80i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.09 - 5.87i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.60 + 4.07i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 + (-4.85 - 3.52i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.85 + 5.70i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.14 + 6.58i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.60 + 4.07i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.14 + 6.58i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.35 - 16.4i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (3.09 + 9.51i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.08269355954783872567357958423, −10.08493398709905127170924807014, −8.945864481323893624627310989325, −8.134371495968616315351794152283, −6.89452567365145465017436322466, −6.48852448395053314727071530874, −5.65072644696463542198352095646, −5.12203835659856003357007531747, −3.48622179568276127522403598019, −1.58075467181620071994366727774, 0.59931033903579789602554468152, 2.66775895681404754907104346364, 3.48096314757935025294999182275, 4.60223451177103911434824791137, 5.38092557217986390100655460436, 6.85507911567487368465384835862, 7.23651637601539957993344636676, 9.228743494971454879763635363191, 9.890011373003245353026307785160, 10.48789822281383344179832801250

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