Properties

Label 2-605-55.32-c1-0-14
Degree $2$
Conductor $605$
Sign $-0.311 - 0.950i$
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  + (−0.895 + 0.895i)2-s + (−1.60 + 1.60i)3-s + 0.396i·4-s + (−0.452 − 2.18i)5-s − 2.87i·6-s + (1.81 − 1.81i)7-s + (−2.14 − 2.14i)8-s − 2.16i·9-s + (2.36 + 1.55i)10-s + (−0.637 − 0.637i)12-s + (3.52 + 3.52i)13-s + 3.24i·14-s + (4.24 + 2.79i)15-s + 3.04·16-s + (0.212 − 0.212i)17-s + (1.93 + 1.93i)18-s + ⋯
L(s)  = 1  + (−0.633 + 0.633i)2-s + (−0.927 + 0.927i)3-s + 0.198i·4-s + (−0.202 − 0.979i)5-s − 1.17i·6-s + (0.684 − 0.684i)7-s + (−0.758 − 0.758i)8-s − 0.721i·9-s + (0.747 + 0.492i)10-s + (−0.184 − 0.184i)12-s + (0.977 + 0.977i)13-s + 0.867i·14-s + (1.09 + 0.721i)15-s + 0.762·16-s + (0.0516 − 0.0516i)17-s + (0.456 + 0.456i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.311 - 0.950i$
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: \((2,\ 605,\ (\ :1/2),\ -0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.437334 + 0.603494i\)
\(L(\frac12)\) \(\approx\) \(0.437334 + 0.603494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.452 + 2.18i)T \)
11 \( 1 \)
good2 \( 1 + (0.895 - 0.895i)T - 2iT^{2} \)
3 \( 1 + (1.60 - 1.60i)T - 3iT^{2} \)
7 \( 1 + (-1.81 + 1.81i)T - 7iT^{2} \)
13 \( 1 + (-3.52 - 3.52i)T + 13iT^{2} \)
17 \( 1 + (-0.212 + 0.212i)T - 17iT^{2} \)
19 \( 1 - 6.08T + 19T^{2} \)
23 \( 1 + (5.07 - 5.07i)T - 23iT^{2} \)
29 \( 1 - 3.98T + 29T^{2} \)
31 \( 1 - 7.59T + 31T^{2} \)
37 \( 1 + (4.56 + 4.56i)T + 37iT^{2} \)
41 \( 1 - 2.42iT - 41T^{2} \)
43 \( 1 + (-1.43 - 1.43i)T + 43iT^{2} \)
47 \( 1 + (-2.49 - 2.49i)T + 47iT^{2} \)
53 \( 1 + (6.21 - 6.21i)T - 53iT^{2} \)
59 \( 1 - 1.17iT - 59T^{2} \)
61 \( 1 + 0.0929iT - 61T^{2} \)
67 \( 1 + (-7.07 - 7.07i)T + 67iT^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + (2.80 + 2.80i)T + 73iT^{2} \)
79 \( 1 - 2.35T + 79T^{2} \)
83 \( 1 + (-5.86 - 5.86i)T + 83iT^{2} \)
89 \( 1 - 2.55iT - 89T^{2} \)
97 \( 1 + (-5.32 - 5.32i)T + 97iT^{2} \)
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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

−10.89816612224380705217089386384, −9.799565520110126796251083685002, −9.251759676186955086915040620016, −8.206315912760862333684146117895, −7.60828289066179452689809422226, −6.38260140199684284383026237097, −5.42303343046220810075847835236, −4.43163741990717340714496617240, −3.74687295958080817380683119331, −1.08947455329126812026733651196, 0.75173023845488469791095698002, 2.02278716144581011276676247807, 3.21022632833904481822272544479, 5.17219371798849772431159074045, 5.98921500217055597935638130608, 6.63055467308709046937595948034, 7.911836473978853349965756583130, 8.490078409554523929874684395551, 9.877191543712623000295329995635, 10.55007043157908215010558411800

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