Properties

Label 2-605-121.16-c1-0-31
Degree $2$
Conductor $605$
Sign $0.605 - 0.795i$
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  + (2.28 + 0.262i)2-s + (1.65 + 1.20i)3-s + (3.22 + 0.750i)4-s + (−0.564 + 0.825i)5-s + (3.48 + 3.19i)6-s + (−1.38 − 1.42i)7-s + (2.84 + 1.01i)8-s + (0.373 + 1.14i)9-s + (−1.50 + 1.74i)10-s + (1.27 + 3.06i)11-s + (4.44 + 5.13i)12-s + (0.362 − 0.601i)13-s + (−2.79 − 3.62i)14-s + (−1.93 + 0.689i)15-s + (0.312 + 0.153i)16-s + (−3.78 + 3.09i)17-s + ⋯
L(s)  = 1  + (1.61 + 0.185i)2-s + (0.958 + 0.696i)3-s + (1.61 + 0.375i)4-s + (−0.252 + 0.369i)5-s + (1.42 + 1.30i)6-s + (−0.523 − 0.538i)7-s + (1.00 + 0.359i)8-s + (0.124 + 0.382i)9-s + (−0.477 + 0.550i)10-s + (0.385 + 0.922i)11-s + (1.28 + 1.48i)12-s + (0.100 − 0.166i)13-s + (−0.747 − 0.969i)14-s + (−0.498 + 0.177i)15-s + (0.0781 + 0.0384i)16-s + (−0.917 + 0.750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.83894 + 1.90280i\)
\(L(\frac12)\) \(\approx\) \(3.83894 + 1.90280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.564 - 0.825i)T \)
11 \( 1 + (-1.27 - 3.06i)T \)
good2 \( 1 + (-2.28 - 0.262i)T + (1.94 + 0.452i)T^{2} \)
3 \( 1 + (-1.65 - 1.20i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (1.38 + 1.42i)T + (-0.199 + 6.99i)T^{2} \)
13 \( 1 + (-0.362 + 0.601i)T + (-6.06 - 11.4i)T^{2} \)
17 \( 1 + (3.78 - 3.09i)T + (3.37 - 16.6i)T^{2} \)
19 \( 1 + (-6.59 - 0.377i)T + (18.8 + 2.16i)T^{2} \)
23 \( 1 + (-1.25 + 8.70i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 + (-1.36 + 3.49i)T + (-21.3 - 19.6i)T^{2} \)
31 \( 1 + (6.33 + 2.67i)T + (21.6 + 22.2i)T^{2} \)
37 \( 1 + (0.532 - 6.19i)T + (-36.4 - 6.30i)T^{2} \)
41 \( 1 + (-4.81 + 2.71i)T + (21.1 - 35.1i)T^{2} \)
43 \( 1 + (8.40 - 2.46i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (0.497 - 2.45i)T + (-43.2 - 18.2i)T^{2} \)
53 \( 1 + (7.36 - 3.61i)T + (32.3 - 41.9i)T^{2} \)
59 \( 1 + (-5.17 - 2.92i)T + (30.4 + 50.5i)T^{2} \)
61 \( 1 + (9.89 - 1.13i)T + (59.4 - 13.8i)T^{2} \)
67 \( 1 + (-2.92 + 6.41i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-2.81 - 10.7i)T + (-61.8 + 34.9i)T^{2} \)
73 \( 1 + (1.29 + 2.45i)T + (-41.2 + 60.2i)T^{2} \)
79 \( 1 + (-0.184 - 6.44i)T + (-78.8 + 4.51i)T^{2} \)
83 \( 1 + (-4.16 - 0.721i)T + (78.1 + 27.8i)T^{2} \)
89 \( 1 + (-12.0 - 7.76i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-6.78 - 9.91i)T + (-35.1 + 90.3i)T^{2} \)
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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

−10.86765413874944568791354457486, −9.951775365045132248287172514197, −9.149292970042953938016203750862, −7.949935835266791374072880597124, −6.89629198641682597243774996421, −6.26907262619288563732482401452, −4.82616239849800926523900607538, −4.08787103328883553183579170365, −3.41029162990582380429951176949, −2.46169285186405541009709533739, 1.75115195353113337680725966093, 3.10234855950294946858697304820, 3.46813739176609511046077855423, 4.98224025081167072192454040995, 5.72563974308409275135402415041, 6.85068483899752381600356600801, 7.60192442406510067592996704434, 8.893539154554120181344888389068, 9.310401444201869410044959885665, 11.12301133644216636363716600145

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