Properties

Label 2-605-11.3-c1-0-34
Degree $2$
Conductor $605$
Sign $-0.999 + 0.0439i$
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  + (1.95 − 1.41i)2-s + (−0.874 − 2.68i)3-s + (1.18 − 3.64i)4-s + (0.809 + 0.587i)5-s + (−5.52 − 4.01i)6-s + (0.618 − 1.90i)7-s + (−1.36 − 4.19i)8-s + (−4.04 + 2.93i)9-s + 2.41·10-s − 10.8·12-s + (−0.947 + 0.688i)13-s + (−1.49 − 4.59i)14-s + (0.874 − 2.68i)15-s + (−2.42 − 1.76i)16-s + (5.52 + 4.01i)17-s + (−3.73 + 11.4i)18-s + ⋯
L(s)  = 1  + (1.38 − 1.00i)2-s + (−0.504 − 1.55i)3-s + (0.591 − 1.82i)4-s + (0.361 + 0.262i)5-s + (−2.25 − 1.63i)6-s + (0.233 − 0.718i)7-s + (−0.482 − 1.48i)8-s + (−1.34 + 0.979i)9-s + 0.763·10-s − 3.12·12-s + (−0.262 + 0.190i)13-s + (−0.398 − 1.22i)14-s + (0.225 − 0.694i)15-s + (−0.606 − 0.440i)16-s + (1.33 + 0.973i)17-s + (−0.879 + 2.70i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.999 + 0.0439i$
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: \((2,\ 605,\ (\ :1/2),\ -0.999 + 0.0439i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0610965 - 2.77847i\)
\(L(\frac12)\) \(\approx\) \(0.0610965 - 2.77847i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 + 1.41i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.874 + 2.68i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.618 + 1.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.947 - 0.688i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.52 - 4.01i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + (-1.13 + 3.47i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.36 - 7.28i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.85 + 5.70i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (-0.874 - 2.68i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.43 - 6.85i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.511 + 1.57i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.53 + 5.47i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 + (9.15 + 6.65i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.362 + 1.11i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.23 + 2.35i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.85 + 3.52i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (2.95 - 2.14i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.65142318295026199332144910575, −9.886544928560333133178779953205, −8.116106854384740686694701638734, −7.30119671691267531254755648238, −6.22930347898547439214780829182, −5.74139466518471457190306096110, −4.54144613373311084393936136669, −3.30850047495763793201893755807, −2.06582526720419860497490719606, −1.19155571365831732829406612797, 2.90424192660060011062777369118, 3.91326797233694192384758085417, 4.90937195285153085844969973365, 5.40154445623970323009605813147, 5.98691769864380659013618842391, 7.26648583809323972309375506366, 8.386210427298731981370624195290, 9.437486846050808058620891821565, 10.13124797049374649105738159640, 11.27953419137247928616079895441

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