Properties

Label 2-605-55.43-c1-0-17
Degree $2$
Conductor $605$
Sign $-0.340 + 0.940i$
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  + (−1.91 − 1.91i)2-s + (0.831 + 0.831i)3-s + 5.35i·4-s + (1.62 − 1.53i)5-s − 3.18i·6-s + (−0.383 − 0.383i)7-s + (6.43 − 6.43i)8-s − 1.61i·9-s + (−6.06 − 0.178i)10-s + (−4.45 + 4.45i)12-s + (1.46 − 1.46i)13-s + 1.46i·14-s + (2.62 + 0.0773i)15-s − 13.9·16-s + (3.33 + 3.33i)17-s + (−3.10 + 3.10i)18-s + ⋯
L(s)  = 1  + (−1.35 − 1.35i)2-s + (0.480 + 0.480i)3-s + 2.67i·4-s + (0.727 − 0.686i)5-s − 1.30i·6-s + (−0.144 − 0.144i)7-s + (2.27 − 2.27i)8-s − 0.539i·9-s + (−1.91 − 0.0563i)10-s + (−1.28 + 1.28i)12-s + (0.405 − 0.405i)13-s + 0.392i·14-s + (0.678 + 0.0199i)15-s − 3.48·16-s + (0.809 + 0.809i)17-s + (−0.730 + 0.730i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.340 + 0.940i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.340 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552960 - 0.788020i\)
\(L(\frac12)\) \(\approx\) \(0.552960 - 0.788020i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.62 + 1.53i)T \)
11 \( 1 \)
good2 \( 1 + (1.91 + 1.91i)T + 2iT^{2} \)
3 \( 1 + (-0.831 - 0.831i)T + 3iT^{2} \)
7 \( 1 + (0.383 + 0.383i)T + 7iT^{2} \)
13 \( 1 + (-1.46 + 1.46i)T - 13iT^{2} \)
17 \( 1 + (-3.33 - 3.33i)T + 17iT^{2} \)
19 \( 1 + 2.45T + 19T^{2} \)
23 \( 1 + (-2.08 - 2.08i)T + 23iT^{2} \)
29 \( 1 + 0.102T + 29T^{2} \)
31 \( 1 + 5.44T + 31T^{2} \)
37 \( 1 + (-5.74 + 5.74i)T - 37iT^{2} \)
41 \( 1 + 3.50iT - 41T^{2} \)
43 \( 1 + (-5.15 + 5.15i)T - 43iT^{2} \)
47 \( 1 + (-1.47 + 1.47i)T - 47iT^{2} \)
53 \( 1 + (0.224 + 0.224i)T + 53iT^{2} \)
59 \( 1 + 2.79iT - 59T^{2} \)
61 \( 1 - 9.29iT - 61T^{2} \)
67 \( 1 + (10.6 - 10.6i)T - 67iT^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + (-6.90 + 6.90i)T - 73iT^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + (-3.83 + 3.83i)T - 83iT^{2} \)
89 \( 1 - 2.37iT - 89T^{2} \)
97 \( 1 + (1.97 - 1.97i)T - 97iT^{2} \)
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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

−10.23583939317213442508968484057, −9.559772265484663246460538091203, −8.894681042559083866271018334125, −8.373100085011502552382172825502, −7.27722619705855403753751102411, −5.81944010651642382502459382424, −4.14513258627748531459946018489, −3.38310758961725195094015932295, −2.14212208311711121798447003413, −0.888438467679521228582067133666, 1.43148178114159590797101847067, 2.65098518272171376409435321694, 4.94214670010689178648550809290, 5.96949173200861314703835502833, 6.66875172151238317236679447031, 7.47352204825817801461058747069, 8.120822022147794534213799115799, 9.127500937019889390886017622663, 9.641935635785622763925578338372, 10.61428826656378178110075096364

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