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Properties

Label 2-605-11.4-c1-0-26
Degree $2$
Conductor $605$
Sign $0.970 - 0.242i$
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  + (1.90 + 1.38i)2-s + (0.885 − 2.72i)3-s + (1.08 + 3.34i)4-s + (0.809 − 0.587i)5-s + (5.44 − 3.95i)6-s + (1.04 + 3.21i)7-s + (−1.10 + 3.38i)8-s + (−4.21 − 3.06i)9-s + 2.34·10-s + 10.0·12-s + (1.19 + 0.871i)13-s + (−2.45 + 7.55i)14-s + (−0.885 − 2.72i)15-s + (−1.08 + 0.785i)16-s + (−3.01 + 2.19i)17-s + (−3.78 − 11.6i)18-s + ⋯
L(s)  = 1  + (1.34 + 0.976i)2-s + (0.511 − 1.57i)3-s + (0.543 + 1.67i)4-s + (0.361 − 0.262i)5-s + (2.22 − 1.61i)6-s + (0.395 + 1.21i)7-s + (−0.389 + 1.19i)8-s + (−1.40 − 1.02i)9-s + 0.742·10-s + 2.91·12-s + (0.332 + 0.241i)13-s + (−0.656 + 2.02i)14-s + (−0.228 − 0.703i)15-s + (−0.270 + 0.196i)16-s + (−0.732 + 0.532i)17-s + (−0.892 − 2.74i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.970 - 0.242i$
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: \((2,\ 605,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.66656 + 0.450717i\)
\(L(\frac12)\) \(\approx\) \(3.66656 + 0.450717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.90 - 1.38i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.885 + 2.72i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-1.04 - 3.21i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.19 - 0.871i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.01 - 2.19i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.993 + 3.05i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.51T + 23T^{2} \)
29 \( 1 + (1.45 + 4.46i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (7.45 + 5.41i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.833 + 2.56i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.40 - 4.32i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.38T + 43T^{2} \)
47 \( 1 + (3.25 - 10.0i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.03 - 1.47i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.68 - 11.3i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (11.0 - 8.04i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.83T + 67T^{2} \)
71 \( 1 + (2.03 - 1.47i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.916 - 2.82i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.47 - 3.97i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.10 + 4.43i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (8.91 + 6.47i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.27764834636393770276424509221, −9.333605138175833047827543380809, −8.496540126973753131882259530497, −7.80542392948154740840426549211, −6.90990568010312411810250783223, −6.08323778334427243817992372727, −5.57659540481148318457948064586, −4.28727914529893494938254422846, −2.80660902112801163439018899173, −1.86247716612169825439855667381, 1.91993562946127223771998850631, 3.43133531859961116046659725146, 3.70284697376768136855742251645, 4.80927945704964615903342630503, 5.33915750769978216646516437116, 6.75952880485145628938966355121, 8.169372412127491527390953412731, 9.326281943482762156865131420878, 10.19322981533803876041014541918, 10.70469733430854149999008068766

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