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Properties

Label 2-605-55.32-c1-0-5
Degree $2$
Conductor $605$
Sign $0.233 + 0.972i$
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  + (−0.875 + 0.875i)2-s + (−1.79 + 1.79i)3-s + 0.466i·4-s + (1.70 + 1.44i)5-s − 3.15i·6-s + (−2.45 + 2.45i)7-s + (−2.15 − 2.15i)8-s − 3.47i·9-s + (−2.75 + 0.235i)10-s + (−0.839 − 0.839i)12-s + (−0.522 − 0.522i)13-s − 4.29i·14-s + (−5.66 + 0.483i)15-s + 2.84·16-s + (0.436 − 0.436i)17-s + (3.04 + 3.04i)18-s + ⋯
L(s)  = 1  + (−0.619 + 0.619i)2-s + (−1.03 + 1.03i)3-s + 0.233i·4-s + (0.764 + 0.644i)5-s − 1.28i·6-s + (−0.926 + 0.926i)7-s + (−0.763 − 0.763i)8-s − 1.15i·9-s + (−0.872 + 0.0744i)10-s + (−0.242 − 0.242i)12-s + (−0.145 − 0.145i)13-s − 1.14i·14-s + (−1.46 + 0.124i)15-s + 0.712·16-s + (0.105 − 0.105i)17-s + (0.716 + 0.716i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.233 + 0.972i$
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: \((2,\ 605,\ (\ :1/2),\ 0.233 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.258016 - 0.203478i\)
\(L(\frac12)\) \(\approx\) \(0.258016 - 0.203478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.70 - 1.44i)T \)
11 \( 1 \)
good2 \( 1 + (0.875 - 0.875i)T - 2iT^{2} \)
3 \( 1 + (1.79 - 1.79i)T - 3iT^{2} \)
7 \( 1 + (2.45 - 2.45i)T - 7iT^{2} \)
13 \( 1 + (0.522 + 0.522i)T + 13iT^{2} \)
17 \( 1 + (-0.436 + 0.436i)T - 17iT^{2} \)
19 \( 1 + 3.14T + 19T^{2} \)
23 \( 1 + (4.30 - 4.30i)T - 23iT^{2} \)
29 \( 1 - 2.90T + 29T^{2} \)
31 \( 1 - 3.03T + 31T^{2} \)
37 \( 1 + (-1.62 - 1.62i)T + 37iT^{2} \)
41 \( 1 + 1.02iT - 41T^{2} \)
43 \( 1 + (-4.07 - 4.07i)T + 43iT^{2} \)
47 \( 1 + (-0.767 - 0.767i)T + 47iT^{2} \)
53 \( 1 + (-3.03 + 3.03i)T - 53iT^{2} \)
59 \( 1 + 7.40iT - 59T^{2} \)
61 \( 1 + 3.74iT - 61T^{2} \)
67 \( 1 + (9.39 + 9.39i)T + 67iT^{2} \)
71 \( 1 - 3.61T + 71T^{2} \)
73 \( 1 + (0.380 + 0.380i)T + 73iT^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 + (10.1 + 10.1i)T + 97iT^{2} \)
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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

−11.14652649243556893430360374980, −10.13288172089842065072780675628, −9.685737306785387515233209638263, −9.008946937068975267853800512986, −7.81452193328622362794955077109, −6.48202980087946989065376746657, −6.17367011695558362952568232633, −5.25078071702680008114388574922, −3.81327190479914985254286361535, −2.68820500284977213758408387449, 0.27218694311179653394125068731, 1.25567532393361161047030212244, 2.45180782865658902892565505551, 4.40130695764236332397986077393, 5.73510216324112850760521132173, 6.23987521635381041154505855278, 7.08038938918960422806258501026, 8.359354245619033326438171307278, 9.301702785109687519563093483712, 10.31445300181943408559546271116

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