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Properties

Label 2-605-55.9-c1-0-32
Degree $2$
Conductor $605$
Sign $0.277 + 0.960i$
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  + (1.17 + 0.381i)2-s + (−0.213 − 0.294i)3-s + (−0.386 − 0.280i)4-s + (−1.33 + 1.79i)5-s + (−0.138 − 0.427i)6-s + (1.51 − 2.09i)7-s + (−1.79 − 2.47i)8-s + (0.886 − 2.72i)9-s + (−2.24 + 1.59i)10-s + 0.173i·12-s + (−2.62 − 0.852i)13-s + (2.58 − 1.87i)14-s + (0.813 + 0.00867i)15-s + (−0.870 − 2.67i)16-s + (3.66 − 1.19i)17-s + (2.07 − 2.86i)18-s + ⋯
L(s)  = 1  + (0.829 + 0.269i)2-s + (−0.123 − 0.170i)3-s + (−0.193 − 0.140i)4-s + (−0.596 + 0.802i)5-s + (−0.0566 − 0.174i)6-s + (0.574 − 0.790i)7-s + (−0.635 − 0.874i)8-s + (0.295 − 0.909i)9-s + (−0.711 + 0.505i)10-s + 0.0501i·12-s + (−0.727 − 0.236i)13-s + (0.689 − 0.501i)14-s + (0.210 + 0.00224i)15-s + (−0.217 − 0.669i)16-s + (0.888 − 0.288i)17-s + (0.490 − 0.674i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24543 - 0.936298i\)
\(L(\frac12)\) \(\approx\) \(1.24543 - 0.936298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.33 - 1.79i)T \)
11 \( 1 \)
good2 \( 1 + (-1.17 - 0.381i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.213 + 0.294i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (-1.51 + 2.09i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (2.62 + 0.852i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.66 + 1.19i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.224 - 0.163i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 8.40iT - 23T^{2} \)
29 \( 1 + (-2.68 - 1.95i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.174 + 0.536i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.307 + 0.422i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.13 + 3.00i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.54iT - 43T^{2} \)
47 \( 1 + (2.89 + 3.98i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (8.29 + 2.69i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.07 - 4.41i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.38 - 13.4i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 3.20iT - 67T^{2} \)
71 \( 1 + (2.59 + 7.98i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.66 - 10.5i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.99 - 9.22i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.13 - 1.01i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 2.48T + 89T^{2} \)
97 \( 1 + (-10.3 - 3.36i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.39729577634580430396387098317, −9.890570980567538181885661633086, −8.604828695837154333806950795014, −7.45182592095486392839919894873, −6.85761778744425583769753616623, −5.95164893358342412175396513382, −4.70969308474962638135688015444, −4.01143064523603403417851773712, −2.96048409652146450565601600119, −0.70436860513408337036462160609, 1.89630556805535016587133519743, 3.31935747790257282777667880065, 4.47734056119236076397725229674, 5.06518911296486356669854096105, 5.75232438967363107237613601145, 7.59884305671828974787903673100, 8.103138037697107455957860991497, 9.042820393993638305957028973539, 9.923250696445255073323423233450, 11.33667144207030159302658826884

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