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Properties

Label 2-605-11.4-c1-0-1
Degree $2$
Conductor $605$
Sign $0.220 - 0.975i$
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  + (−1.51 − 1.10i)2-s + (−0.549 + 1.69i)3-s + (0.469 + 1.44i)4-s + (−0.809 + 0.587i)5-s + (2.69 − 1.96i)6-s + (−1.31 − 4.04i)7-s + (−0.278 + 0.857i)8-s + (−0.128 − 0.0937i)9-s + 1.87·10-s − 2.70·12-s + (1.10 + 0.799i)13-s + (−2.46 + 7.59i)14-s + (−0.549 − 1.69i)15-s + (3.82 − 2.78i)16-s + (1.69 − 1.23i)17-s + (0.0924 + 0.284i)18-s + ⋯
L(s)  = 1  + (−1.07 − 0.779i)2-s + (−0.317 + 0.976i)3-s + (0.234 + 0.722i)4-s + (−0.361 + 0.262i)5-s + (1.10 − 0.800i)6-s + (−0.497 − 1.53i)7-s + (−0.0984 + 0.303i)8-s + (−0.0429 − 0.0312i)9-s + 0.593·10-s − 0.779·12-s + (0.305 + 0.221i)13-s + (−0.659 + 2.03i)14-s + (−0.141 − 0.436i)15-s + (0.956 − 0.695i)16-s + (0.411 − 0.299i)17-s + (0.0217 + 0.0670i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.220 - 0.975i$
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: \((2,\ 605,\ (\ :1/2),\ 0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.316050 + 0.252497i\)
\(L(\frac12)\) \(\approx\) \(0.316050 + 0.252497i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 + 1.10i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.549 - 1.69i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (1.31 + 4.04i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.10 - 0.799i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.69 + 1.23i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.186 - 0.574i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.39T + 23T^{2} \)
29 \( 1 + (-2.04 - 6.30i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.77 - 1.29i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.90 - 5.86i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.28 - 7.04i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 + (0.950 - 2.92i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.38 - 3.91i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.71 - 11.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.60 - 3.34i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.86T + 67T^{2} \)
71 \( 1 + (-4.23 + 3.07i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.223 - 0.687i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.58 - 3.33i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.770 - 0.559i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.24T + 89T^{2} \)
97 \( 1 + (9.33 + 6.78i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.48708574993987208580655835993, −10.16757361040998571015894816672, −9.557518466766824378690608567301, −8.410325278991600125779367175197, −7.54316208386551668891819260937, −6.54224538120656466357604626568, −5.05502489565496214035296483507, −4.02225814085939091170024524760, −3.14199008653502902287572930334, −1.25263655192379891837087500494, 0.37546908003625031466850251314, 2.02716889362382925814886590079, 3.67682157334506313894818887289, 5.50999884385311321276209730560, 6.23872853303497262241274597767, 6.91241486346132054264795658399, 8.052859978726141338152429751974, 8.374842636579987368610871945850, 9.414244751006189869937684414917, 10.05132174233566045140507885075

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