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Properties

Label 2-605-55.49-c1-0-42
Degree $2$
Conductor $605$
Sign $0.202 + 0.979i$
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  + (1.92 − 0.625i)2-s + (1.54 − 2.12i)3-s + (1.69 − 1.23i)4-s + (2.01 + 0.965i)5-s + (1.63 − 5.04i)6-s + (−0.567 − 0.781i)7-s + (0.113 − 0.156i)8-s + (−1.19 − 3.67i)9-s + (4.48 + 0.596i)10-s − 5.49i·12-s + (−4.30 + 1.39i)13-s + (−1.58 − 1.14i)14-s + (5.15 − 2.78i)15-s + (−1.17 + 3.61i)16-s + (−3.17 − 1.03i)17-s + (−4.60 − 6.33i)18-s + ⋯
L(s)  = 1  + (1.36 − 0.442i)2-s + (0.889 − 1.22i)3-s + (0.847 − 0.615i)4-s + (0.901 + 0.431i)5-s + (0.669 − 2.05i)6-s + (−0.214 − 0.295i)7-s + (0.0400 − 0.0551i)8-s + (−0.398 − 1.22i)9-s + (1.41 + 0.188i)10-s − 1.58i·12-s + (−1.19 + 0.387i)13-s + (−0.422 − 0.307i)14-s + (1.33 − 0.720i)15-s + (−0.293 + 0.903i)16-s + (−0.769 − 0.250i)17-s + (−1.08 − 1.49i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.202 + 0.979i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.202 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.11715 - 2.53868i\)
\(L(\frac12)\) \(\approx\) \(3.11715 - 2.53868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.01 - 0.965i)T \)
11 \( 1 \)
good2 \( 1 + (-1.92 + 0.625i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.54 + 2.12i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (0.567 + 0.781i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (4.30 - 1.39i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.17 + 1.03i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.65 - 1.92i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.36iT - 23T^{2} \)
29 \( 1 + (-3.97 + 2.88i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.129 + 0.397i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.72 + 5.12i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.68 + 3.40i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.26iT - 43T^{2} \)
47 \( 1 + (-2.54 + 3.49i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.53 - 0.822i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.19 - 5.95i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.763 - 2.34i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 9.60iT - 67T^{2} \)
71 \( 1 + (1.68 - 5.18i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.843 - 1.16i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.310 + 0.954i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (7.03 + 2.28i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-2.87 + 0.932i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.63004538742188053183108089241, −9.618725086194891200741064080529, −8.713588699179914952058642711087, −7.44146184072557206408352726075, −6.82877151541152471662922583091, −5.89387755255201938326455004820, −4.82698910662424369317287833055, −3.45979239420103907356077823574, −2.54294343402905140654200811401, −1.83291250549206385350278451608, 2.52435962631446559176119121830, 3.26320594237124508695180975796, 4.69897887964560934275537188973, 4.83826435382904225666728188708, 5.99959948361916572115710539415, 6.98841340946696650230043624004, 8.389886816740076369163427535899, 9.241462570516649487820228158258, 9.817561276654369810644131691786, 10.63046754383201847878201845856

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