Properties

Label 2-605-11.5-c1-0-8
Degree $2$
Conductor $605$
Sign $0.882 - 0.469i$
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  + (0.127 + 0.393i)2-s + (−2.28 − 1.66i)3-s + (1.47 − 1.07i)4-s + (−0.309 + 0.951i)5-s + (0.362 − 1.11i)6-s + (−1.61 + 1.17i)7-s + (1.28 + 0.932i)8-s + (1.54 + 4.75i)9-s − 0.414·10-s − 5.17·12-s + (2.11 + 6.49i)13-s + (−0.670 − 0.486i)14-s + (2.28 − 1.66i)15-s + (0.927 − 2.85i)16-s + (−0.362 + 1.11i)17-s + (−1.67 + 1.21i)18-s + ⋯
L(s)  = 1  + (0.0905 + 0.278i)2-s + (−1.32 − 0.959i)3-s + (0.739 − 0.537i)4-s + (−0.138 + 0.425i)5-s + (0.147 − 0.454i)6-s + (−0.611 + 0.444i)7-s + (0.453 + 0.329i)8-s + (0.515 + 1.58i)9-s − 0.130·10-s − 1.49·12-s + (0.585 + 1.80i)13-s + (−0.179 − 0.130i)14-s + (0.590 − 0.429i)15-s + (0.231 − 0.713i)16-s + (−0.0878 + 0.270i)17-s + (−0.394 + 0.286i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.882 - 0.469i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.882 - 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02486 + 0.255832i\)
\(L(\frac12)\) \(\approx\) \(1.02486 + 0.255832i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.127 - 0.393i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (2.28 + 1.66i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (1.61 - 1.17i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-2.11 - 6.49i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.362 - 1.11i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + (-6.19 + 4.50i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.95 - 2.14i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.85 - 3.52i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (-2.28 - 1.66i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.106 - 0.326i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.81 + 5.67i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.11 - 12.6i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 4.48T + 67T^{2} \)
71 \( 1 + (3.49 - 10.7i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.52 - 4.01i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.23 + 3.80i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.85 + 5.70i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 9.31T + 89T^{2} \)
97 \( 1 + (2.36 + 7.28i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.99284793698350527813499150318, −10.15021625576368863822845465056, −8.976473233524230955904288697341, −7.59208254103792757710015168944, −6.75358635196571883653271200906, −6.36422436304158331162700240688, −5.69955885786080111162698146582, −4.43464313202521559223413210483, −2.53473080909625189039432969968, −1.30301107886333873325601333491, 0.75639729838863468353415584226, 3.06936642421916171185677208465, 3.92349245121681516457866480384, 5.04228174135960546839921953981, 5.91357303320321488455808145183, 6.82026141146853880411811550371, 7.85087151347450406111493244860, 9.020261380785152340016355010968, 10.25652977586491593236120269116, 10.58014824704398885086092655919

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