Properties

Label 2-605-55.32-c1-0-6
Degree $2$
Conductor $605$
Sign $-0.539 - 0.841i$
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  + (−1.05 + 1.05i)2-s + (1.53 − 1.53i)3-s − 0.238i·4-s + (−1.92 − 1.13i)5-s + 3.25i·6-s + (−1.66 + 1.66i)7-s + (−1.86 − 1.86i)8-s − 1.73i·9-s + (3.23 − 0.835i)10-s + (−0.366 − 0.366i)12-s + (4.40 + 4.40i)13-s − 3.52i·14-s + (−4.71 + 1.21i)15-s + 4.41·16-s + (−3.57 + 3.57i)17-s + (1.83 + 1.83i)18-s + ⋯
L(s)  = 1  + (−0.748 + 0.748i)2-s + (0.888 − 0.888i)3-s − 0.119i·4-s + (−0.861 − 0.508i)5-s + 1.32i·6-s + (−0.629 + 0.629i)7-s + (−0.658 − 0.658i)8-s − 0.578i·9-s + (1.02 − 0.264i)10-s + (−0.105 − 0.105i)12-s + (1.22 + 1.22i)13-s − 0.941i·14-s + (−1.21 + 0.313i)15-s + 1.10·16-s + (−0.867 + 0.867i)17-s + (0.432 + 0.432i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.539 - 0.841i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.539 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342284 + 0.625971i\)
\(L(\frac12)\) \(\approx\) \(0.342284 + 0.625971i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.92 + 1.13i)T \)
11 \( 1 \)
good2 \( 1 + (1.05 - 1.05i)T - 2iT^{2} \)
3 \( 1 + (-1.53 + 1.53i)T - 3iT^{2} \)
7 \( 1 + (1.66 - 1.66i)T - 7iT^{2} \)
13 \( 1 + (-4.40 - 4.40i)T + 13iT^{2} \)
17 \( 1 + (3.57 - 3.57i)T - 17iT^{2} \)
19 \( 1 + 0.201T + 19T^{2} \)
23 \( 1 + (2.77 - 2.77i)T - 23iT^{2} \)
29 \( 1 + 2.19T + 29T^{2} \)
31 \( 1 + 6.45T + 31T^{2} \)
37 \( 1 + (-6.07 - 6.07i)T + 37iT^{2} \)
41 \( 1 - 2.91iT - 41T^{2} \)
43 \( 1 + (2.16 + 2.16i)T + 43iT^{2} \)
47 \( 1 + (-6.27 - 6.27i)T + 47iT^{2} \)
53 \( 1 + (5.53 - 5.53i)T - 53iT^{2} \)
59 \( 1 + 10.9iT - 59T^{2} \)
61 \( 1 + 3.92iT - 61T^{2} \)
67 \( 1 + (0.637 + 0.637i)T + 67iT^{2} \)
71 \( 1 - 1.45T + 71T^{2} \)
73 \( 1 + (6.80 + 6.80i)T + 73iT^{2} \)
79 \( 1 + 8.04T + 79T^{2} \)
83 \( 1 + (-1.70 - 1.70i)T + 83iT^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + (2.07 + 2.07i)T + 97iT^{2} \)
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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

−11.02277128843036925706521744622, −9.344274005143307113218116485569, −8.974614897214383622544701782325, −8.265765973252134865914423735132, −7.66474418469989748068468758019, −6.72506711316944004289019860040, −6.04254659097467947578514663961, −4.16392036108595142788434465169, −3.21166671571148599259824674257, −1.63752961882235106765286256350, 0.45739721455470797714045643655, 2.58976002009158631404546167370, 3.43860706530923406740125876207, 4.16684019540154055524713794460, 5.77294992371514601971061396872, 7.03021217825965320253991252627, 8.140596577174149245915891390486, 8.795744049036367785408489299701, 9.571963929504042270088907779100, 10.43496149042168182601422496754

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