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Properties

Label 2-605-55.9-c1-0-27
Degree $2$
Conductor $605$
Sign $-0.649 + 0.760i$
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  + (−1.83 − 0.596i)2-s + (0.304 + 0.418i)3-s + (1.40 + 1.01i)4-s + (0.809 − 2.08i)5-s + (−0.309 − 0.951i)6-s + (−0.526 + 0.725i)7-s + (0.304 + 0.418i)8-s + (0.844 − 2.59i)9-s + (−2.73 + 3.34i)10-s + 0.896i·12-s + (−4.03 − 1.31i)13-s + (1.40 − 1.01i)14-s + (1.11 − 0.295i)15-s + (−1.37 − 4.24i)16-s + (0.984 − 0.319i)17-s + (−3.10 + 4.26i)18-s + ⋯
L(s)  = 1  + (−1.29 − 0.422i)2-s + (0.175 + 0.241i)3-s + (0.700 + 0.509i)4-s + (0.362 − 0.932i)5-s + (−0.126 − 0.388i)6-s + (−0.199 + 0.274i)7-s + (0.107 + 0.148i)8-s + (0.281 − 0.866i)9-s + (−0.863 + 1.05i)10-s + 0.258i·12-s + (−1.11 − 0.363i)13-s + (0.374 − 0.272i)14-s + (0.288 − 0.0761i)15-s + (−0.344 − 1.06i)16-s + (0.238 − 0.0775i)17-s + (−0.731 + 1.00i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.649 + 0.760i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262246 - 0.568741i\)
\(L(\frac12)\) \(\approx\) \(0.262246 - 0.568741i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 + 2.08i)T \)
11 \( 1 \)
good2 \( 1 + (1.83 + 0.596i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.304 - 0.418i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (0.526 - 0.725i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (4.03 + 1.31i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.984 + 0.319i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.01 + 3.64i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 6.31iT - 23T^{2} \)
29 \( 1 + (5.60 + 4.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.62 - 5.01i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.93 + 5.41i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.40 + 1.01i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 + (2.41 + 3.32i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (11.3 + 3.69i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.82 + 2.78i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.55 + 7.86i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 14.9iT - 67T^{2} \)
71 \( 1 + (-0.678 - 2.08i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.87 - 3.96i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.68 + 5.19i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (9.41 - 3.05i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 6.46T + 89T^{2} \)
97 \( 1 + (-0.624 - 0.202i)T + (78.4 + 57.0i)T^{2} \)
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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

−9.821073141400830015090922974860, −9.474323592873204447702783113399, −9.054795936707141307122346161688, −7.899612222102100861217056815559, −7.18751265241157978842038023182, −5.65472895783571363215675478489, −4.82542346566794996422449189107, −3.28851367211754536033390861118, −1.89633002984233114885036533400, −0.53179777966070140744448061264, 1.62161858143216239269087376701, 2.88866077327227936207459266568, 4.45455541465408682510031414940, 5.90446252047374509435498452006, 6.95233907131603796553161087466, 7.50600656740360869946172499206, 8.119834347475823569121500527008, 9.429774008771561549482244548383, 9.904641412768198370044734102531, 10.58925620751378528373527881400

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