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Properties

Label 2-605-11.5-c1-0-2
Degree $2$
Conductor $605$
Sign $0.0219 + 0.999i$
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  + (0.848 + 2.61i)2-s + (0.170 + 0.123i)3-s + (−4.47 + 3.25i)4-s + (−0.309 + 0.951i)5-s + (−0.178 + 0.550i)6-s + (−1.87 + 1.36i)7-s + (−7.84 − 5.70i)8-s + (−0.913 − 2.81i)9-s − 2.74·10-s − 1.16·12-s + (0.165 + 0.507i)13-s + (−5.15 − 3.74i)14-s + (−0.170 + 0.123i)15-s + (4.80 − 14.7i)16-s + (−0.748 + 2.30i)17-s + (6.56 − 4.76i)18-s + ⋯
L(s)  = 1  + (0.599 + 1.84i)2-s + (0.0984 + 0.0715i)3-s + (−2.23 + 1.62i)4-s + (−0.138 + 0.425i)5-s + (−0.0729 + 0.224i)6-s + (−0.710 + 0.516i)7-s + (−2.77 − 2.01i)8-s + (−0.304 − 0.936i)9-s − 0.867·10-s − 0.336·12-s + (0.0457 + 0.140i)13-s + (−1.37 − 1.00i)14-s + (−0.0440 + 0.0319i)15-s + (1.20 − 3.69i)16-s + (−0.181 + 0.558i)17-s + (1.54 − 1.12i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.0219 + 0.999i$
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: \((2,\ 605,\ (\ :1/2),\ 0.0219 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.624764 - 0.611176i\)
\(L(\frac12)\) \(\approx\) \(0.624764 - 0.611176i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.848 - 2.61i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.170 - 0.123i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (1.87 - 1.36i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.165 - 0.507i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.748 - 2.30i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.00 - 2.91i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.53T + 23T^{2} \)
29 \( 1 + (4.44 - 3.22i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.322 - 0.993i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.05 - 4.40i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-8.57 - 6.23i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.32T + 43T^{2} \)
47 \( 1 + (5.47 + 3.97i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.40 + 4.31i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.21 + 3.78i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.10 - 6.46i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 0.721T + 67T^{2} \)
71 \( 1 + (-1.40 + 4.31i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.864 + 0.627i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.43 + 4.41i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.20 - 12.9i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (-1.47 - 4.52i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.69051910507139004127323525918, −9.965120778215570732410500819731, −9.261714543684907914273318925121, −8.441335503522705728749397518157, −7.60366540558954291801302822468, −6.56831810208393613735788362824, −6.14240725192145995958091417264, −5.22010824869154484551042658189, −3.86073511918167986425639319259, −3.23642956330706207767427032728, 0.38858492215646682007864731278, 1.97953740234973938547466762769, 3.06328254784705293438106030270, 4.03246175718473033491834705417, 4.98585035728919940912915860149, 5.82707485744800697123940074114, 7.46903306041060380381885621991, 8.669674134135742533114671725276, 9.520706286088779670584545928662, 10.14750535697643621040022085230

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