Properties

Label 2-605-11.9-c1-0-30
Degree $2$
Conductor $605$
Sign $0.263 + 0.964i$
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  + (0.227 − 0.701i)2-s + (2.27 − 1.65i)3-s + (1.17 + 0.855i)4-s + (−0.309 − 0.951i)5-s + (−0.640 − 1.97i)6-s + (0.834 + 0.606i)7-s + (2.06 − 1.49i)8-s + (1.51 − 4.66i)9-s − 0.737·10-s + 4.09·12-s + (−1.06 + 3.28i)13-s + (0.615 − 0.447i)14-s + (−2.27 − 1.65i)15-s + (0.318 + 0.980i)16-s + (−0.741 − 2.28i)17-s + (−2.92 − 2.12i)18-s + ⋯
L(s)  = 1  + (0.161 − 0.496i)2-s + (1.31 − 0.954i)3-s + (0.588 + 0.427i)4-s + (−0.138 − 0.425i)5-s + (−0.261 − 0.805i)6-s + (0.315 + 0.229i)7-s + (0.729 − 0.529i)8-s + (0.505 − 1.55i)9-s − 0.233·10-s + 1.18·12-s + (−0.295 + 0.909i)13-s + (0.164 − 0.119i)14-s + (−0.587 − 0.426i)15-s + (0.0796 + 0.245i)16-s + (−0.179 − 0.553i)17-s + (−0.689 − 0.501i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23481 - 1.70630i\)
\(L(\frac12)\) \(\approx\) \(2.23481 - 1.70630i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.227 + 0.701i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-2.27 + 1.65i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (-0.834 - 0.606i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.06 - 3.28i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.741 + 2.28i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (6.20 - 4.50i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 2.45T + 23T^{2} \)
29 \( 1 + (4.81 + 3.49i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.13 - 3.50i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.82 + 3.50i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.18 + 2.31i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 7.64T + 43T^{2} \)
47 \( 1 + (4.72 - 3.43i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.66 - 11.2i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.38 + 1.73i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.766 + 2.35i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 6.14T + 67T^{2} \)
71 \( 1 + (-0.625 - 1.92i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.668 + 0.485i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.73 + 11.4i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.497 + 1.53i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.16T + 89T^{2} \)
97 \( 1 + (-0.754 + 2.32i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.61262218392199821905842002252, −9.304336628269636350587607554686, −8.678518977277810644139597230504, −7.77128122089826253828295989971, −7.22481180422378377399439875988, −6.21629397636747407530279693303, −4.46850743641937402148457602038, −3.48719627724127720599627646063, −2.30892045818043559697041177471, −1.67149842358766245229019965562, 2.10759082469302385646708992405, 3.08420508643665548196841873983, 4.26039784723613228593367360503, 5.16984265397148104691433455799, 6.43003955635254612594898203474, 7.43499149613848193034634161573, 8.147785099676351323500639834506, 9.015123571813916440059389304783, 10.01732808871321403869581535385, 10.74022533420700226304134559975

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