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Properties

Label 2-605-55.28-c1-0-16
Degree $2$
Conductor $605$
Sign $0.951 - 0.308i$
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  + (0.261 − 0.513i)2-s + (0.120 + 0.760i)3-s + (0.980 + 1.34i)4-s + (−1.76 − 1.36i)5-s + (0.421 + 0.136i)6-s + (−1.17 − 0.186i)7-s + (2.08 − 0.330i)8-s + (2.28 − 0.743i)9-s + (−1.16 + 0.550i)10-s + (−0.908 + 0.908i)12-s + (5.41 + 2.75i)13-s + (−0.403 + 0.555i)14-s + (0.825 − 1.51i)15-s + (−0.655 + 2.01i)16-s + (−0.330 + 0.168i)17-s + (0.216 − 1.36i)18-s + ⋯
L(s)  = 1  + (0.184 − 0.362i)2-s + (0.0695 + 0.438i)3-s + (0.490 + 0.674i)4-s + (−0.791 − 0.611i)5-s + (0.172 + 0.0559i)6-s + (−0.445 − 0.0705i)7-s + (0.737 − 0.116i)8-s + (0.763 − 0.247i)9-s + (−0.368 + 0.174i)10-s + (−0.262 + 0.262i)12-s + (1.50 + 0.765i)13-s + (−0.107 + 0.148i)14-s + (0.213 − 0.389i)15-s + (−0.163 + 0.504i)16-s + (−0.0801 + 0.0408i)17-s + (0.0511 − 0.322i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.951 - 0.308i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (578, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.951 - 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77895 + 0.281551i\)
\(L(\frac12)\) \(\approx\) \(1.77895 + 0.281551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.76 + 1.36i)T \)
11 \( 1 \)
good2 \( 1 + (-0.261 + 0.513i)T + (-1.17 - 1.61i)T^{2} \)
3 \( 1 + (-0.120 - 0.760i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (1.17 + 0.186i)T + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (-5.41 - 2.75i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.330 - 0.168i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (1.11 + 0.809i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-3.48 - 3.48i)T + 23iT^{2} \)
29 \( 1 + (-4.95 + 3.60i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.764 + 2.35i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.465 - 2.93i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-3.60 + 4.95i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (6.75 - 6.75i)T - 43iT^{2} \)
47 \( 1 + (1.26 - 0.199i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-0.185 + 0.363i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-0.110 - 0.151i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.649 - 0.211i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-7.14 + 7.14i)T - 67iT^{2} \)
71 \( 1 + (-0.319 + 0.983i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.715 - 4.51i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (3.59 + 11.0i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.16 + 10.1i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 + 8.04iT - 89T^{2} \)
97 \( 1 + (-7.52 - 3.83i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.97418625777732649296425126719, −9.853386289900283600124331333047, −8.913409835066150016884988941926, −8.156320476060265301622362605423, −7.14082022683228942145844492779, −6.34717165660335893102914845725, −4.69124171530326764603027440596, −3.94947789274649404874707922637, −3.24689745405073286260337236044, −1.44435103743925644047187360836, 1.16540166061037276910776422388, 2.75131383075184411701353612072, 3.96398674819113832174266501527, 5.20439252027937102231662190305, 6.48212859485467857888736789051, 6.76069312384855901575906855249, 7.79819866967599093785084486003, 8.587972575281850960093806710862, 10.03578731329574560856808905949, 10.67766431004574645072018803698

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