Properties

Label 2-605-11.5-c1-0-25
Degree $2$
Conductor $605$
Sign $-0.0219 - 0.999i$
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  + (0.579 + 1.78i)2-s + (1.43 + 1.04i)3-s + (−1.22 + 0.893i)4-s + (0.309 − 0.951i)5-s + (−1.03 + 3.17i)6-s + (3.44 − 2.50i)7-s + (0.729 + 0.529i)8-s + (0.0492 + 0.151i)9-s + 1.87·10-s − 2.70·12-s + (−0.420 − 1.29i)13-s + (6.46 + 4.69i)14-s + (1.43 − 1.04i)15-s + (−1.46 + 4.49i)16-s + (−0.648 + 1.99i)17-s + (−0.242 + 0.175i)18-s + ⋯
L(s)  = 1  + (0.409 + 1.26i)2-s + (0.830 + 0.603i)3-s + (−0.614 + 0.446i)4-s + (0.138 − 0.425i)5-s + (−0.420 + 1.29i)6-s + (1.30 − 0.945i)7-s + (0.257 + 0.187i)8-s + (0.0164 + 0.0505i)9-s + 0.593·10-s − 0.779·12-s + (−0.116 − 0.358i)13-s + (1.72 + 1.25i)14-s + (0.371 − 0.269i)15-s + (−0.365 + 1.12i)16-s + (−0.157 + 0.484i)17-s + (−0.0570 + 0.0414i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0219 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.0219 - 0.999i$
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: \((2,\ 605,\ (\ :1/2),\ -0.0219 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92738 + 1.97023i\)
\(L(\frac12)\) \(\approx\) \(1.92738 + 1.97023i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.579 - 1.78i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.43 - 1.04i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-3.44 + 2.50i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.420 + 1.29i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.648 - 1.99i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.489 - 0.355i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.39T + 23T^{2} \)
29 \( 1 + (5.36 - 3.89i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.678 + 2.08i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.99 - 3.62i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.99 - 4.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 + (-2.48 - 1.80i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.05 + 6.32i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (9.73 - 7.07i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.75 + 5.41i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.86T + 67T^{2} \)
71 \( 1 + (1.61 - 4.98i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.584 - 0.424i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.75 + 5.39i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.294 + 0.906i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 1.24T + 89T^{2} \)
97 \( 1 + (-3.56 - 10.9i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.71781193127602514184608232858, −9.884839041709828678480883720388, −8.734711688998510813250172416799, −8.105247718977717087753632726623, −7.51732344897369734933992158663, −6.36776321787247425556417828890, −5.25316210604702425026416479048, −4.49720887092735302177943362325, −3.66520155658123024478327739909, −1.75295511597219884817295510815, 1.83702088180524767514776469097, 2.20579653561064809224983340824, 3.31409900325720910799422227417, 4.58052337342795661363555266480, 5.58336347000763909373715865397, 7.10760485565435548042948980241, 7.86942152167799877640588282525, 8.758293955520109390025921047168, 9.636808385541091223368301848467, 10.75488110164476792009667165250

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