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Properties

Label 2-605-55.4-c1-0-45
Degree $2$
Conductor $605$
Sign $-0.530 - 0.847i$
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  + (1.46 − 2.02i)2-s + (−1.30 − 4.02i)4-s + (−1.80 + 1.31i)5-s + (−2.93 + 0.953i)7-s + (−5.31 − 1.72i)8-s + (−2.42 − 1.76i)9-s + 5.58i·10-s + (−1.12 + 1.54i)13-s + (−2.38 + 7.33i)14-s + (−4.42 + 3.21i)16-s + (−4.74 − 6.53i)17-s + (−7.12 + 2.31i)18-s + (7.66 + 5.56i)20-s + (1.54 − 4.75i)25-s + (1.47 + 4.53i)26-s + ⋯
L(s)  = 1  + (1.03 − 1.42i)2-s + (−0.654 − 2.01i)4-s + (−0.809 + 0.587i)5-s + (−1.10 + 0.360i)7-s + (−1.87 − 0.610i)8-s + (−0.809 − 0.587i)9-s + 1.76i·10-s + (−0.310 + 0.428i)13-s + (−0.636 + 1.95i)14-s + (−1.10 + 0.804i)16-s + (−1.15 − 1.58i)17-s + (−1.67 + 0.545i)18-s + (1.71 + 1.24i)20-s + (0.309 − 0.951i)25-s + (0.288 + 0.888i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.530 - 0.847i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.352112 + 0.635436i\)
\(L(\frac12)\) \(\approx\) \(0.352112 + 0.635436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.80 - 1.31i)T \)
11 \( 1 \)
good2 \( 1 + (-1.46 + 2.02i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (2.93 - 0.953i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.12 - 1.54i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.74 + 6.53i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.23 - 5.25i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 13.0iT - 43T^{2} \)
47 \( 1 + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.23 + 3.80i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (6.47 - 4.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (11.3 - 3.67i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.428 - 0.589i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + (-29.9 - 92.2i)T^{2} \)
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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

−10.36735136092377297352871834075, −9.484004351743871256346069687482, −8.733471132213580223641843326348, −7.04218558525261981468590535644, −6.30103862113117905312487713629, −5.09765959005406587258497438116, −4.06733012926720557251421115441, −3.07523112776643168439152179468, −2.57063137206641762348552611520, −0.27026797428729992857328065727, 3.00535894091498237184184576529, 4.05651294628167799671206376295, 4.77333600248184704822200630309, 5.94405035793679060441491329677, 6.51240018802840265143258427533, 7.66686944612933737310802292228, 8.196569241524542132263658351947, 9.038332711501303689060017883960, 10.38979221314028170092558484235, 11.49718898710246098560964113321

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