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Properties

Label 2-605-11.4-c1-0-15
Degree $2$
Conductor $605$
Sign $0.0915 - 0.995i$
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  + (1.40 + 1.01i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (−1.40 + 1.01i)6-s + (0.535 + 1.64i)7-s + (0.535 − 1.64i)8-s + (1.61 + 1.17i)9-s + 1.73·10-s − 0.999·12-s + (2.80 + 2.03i)13-s + (−0.927 + 2.85i)14-s + (0.309 + 0.951i)15-s + (4.04 − 2.93i)16-s + (−5.60 + 4.07i)17-s + (1.07 + 3.29i)18-s + ⋯
L(s)  = 1  + (0.990 + 0.719i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.572 + 0.415i)6-s + (0.202 + 0.622i)7-s + (0.189 − 0.582i)8-s + (0.539 + 0.391i)9-s + 0.547·10-s − 0.288·12-s + (0.777 + 0.564i)13-s + (−0.247 + 0.762i)14-s + (0.0797 + 0.245i)15-s + (1.01 − 0.734i)16-s + (−1.35 + 0.987i)17-s + (0.252 + 0.776i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0915 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0915 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.0915 - 0.995i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.0915 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91763 + 1.74947i\)
\(L(\frac12)\) \(\approx\) \(1.91763 + 1.74947i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 - 1.01i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.309 - 0.951i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.535 - 1.64i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.80 - 2.03i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.60 - 4.07i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.07 + 3.29i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-6.47 - 4.70i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.47 + 7.60i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.74 - 11.5i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.66T + 43T^{2} \)
47 \( 1 + (-2.78 + 8.55i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.85 + 3.52i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.70 + 11.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-7.00 + 5.09i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 5T + 67T^{2} \)
71 \( 1 + (-9.70 + 7.05i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.40 + 6.10i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.80 + 2.03i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-8.09 - 5.87i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.85642810533865463184113848547, −10.01740219055993922275251503439, −9.074954574487397878846748815409, −8.226540034887620095146825691884, −6.81928830863264461319132411558, −6.27915288890610616927387786932, −5.13662178126344212933699884746, −4.65481537137729941675002513725, −3.63277801244751773552413055356, −1.83587833855926748320594607384, 1.30708549762114322813216733994, 2.60296037687386123714273268574, 3.77306855096086621822495083912, 4.62270013278969348542253652133, 5.78790460450726178708113176332, 6.70399622563410660018903436009, 7.63131211312399217636785859046, 8.659323662254706107942792269375, 9.941667834838351301819564831343, 10.68046990399416354771957533270

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