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Properties

Label 2-605-55.4-c1-0-11
Degree $2$
Conductor $605$
Sign $-0.368 - 0.929i$
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  + (−1.18 + 1.63i)2-s + (−2.49 − 0.809i)3-s + (−0.647 − 1.99i)4-s + (1.54 + 1.61i)5-s + (4.29 − 3.11i)6-s + (−0.918 + 0.298i)7-s + (0.183 + 0.0595i)8-s + (3.12 + 2.27i)9-s + (−4.48 + 0.596i)10-s + 5.49i·12-s + (2.65 − 3.66i)13-s + (0.603 − 1.85i)14-s + (−2.53 − 5.28i)15-s + (3.07 − 2.23i)16-s + (1.96 + 2.69i)17-s + (−7.44 + 2.41i)18-s + ⋯
L(s)  = 1  + (−0.841 + 1.15i)2-s + (−1.43 − 0.467i)3-s + (−0.323 − 0.996i)4-s + (0.689 + 0.724i)5-s + (1.75 − 1.27i)6-s + (−0.347 + 0.112i)7-s + (0.0648 + 0.0210i)8-s + (1.04 + 0.757i)9-s + (−1.41 + 0.188i)10-s + 1.58i·12-s + (0.737 − 1.01i)13-s + (0.161 − 0.496i)14-s + (−0.653 − 1.36i)15-s + (0.768 − 0.558i)16-s + (0.475 + 0.654i)17-s + (−1.75 + 0.570i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.368 - 0.929i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.368 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.315121 + 0.463652i\)
\(L(\frac12)\) \(\approx\) \(0.315121 + 0.463652i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.54 - 1.61i)T \)
11 \( 1 \)
good2 \( 1 + (1.18 - 1.63i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (2.49 + 0.809i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.918 - 0.298i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.65 + 3.66i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.96 - 2.69i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.01 + 3.11i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 3.36iT - 23T^{2} \)
29 \( 1 + (-1.51 - 4.67i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.338 - 0.245i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-6.02 + 1.95i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.78 - 5.50i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.26iT - 43T^{2} \)
47 \( 1 + (4.11 + 1.33i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.56 - 2.15i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.12 - 9.62i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.99 - 1.45i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.60iT - 67T^{2} \)
71 \( 1 + (-4.41 + 3.20i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.36 + 0.443i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.812 + 0.590i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.34 - 5.98i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-1.77 + 2.44i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.67605313879892468755844930305, −10.13947849137302254827205345300, −9.112968111903066252998506381758, −8.041812204858458877450814439208, −7.13872800598684659099324355085, −6.33034845426982394046010062034, −6.00968086261727005899338223293, −5.12591644500853002719667774686, −3.08396043688060073612343170100, −1.00747074389092170506248316985, 0.66347527096791007356243182905, 1.85538983489232495567484363773, 3.59258039940966966008400033405, 4.80026134010246212275929193047, 5.79722148200468888913775590530, 6.47108085727626479011779302368, 8.078181893842810749801810047307, 9.195974439716780755080453118295, 9.754673582585277480259435204794, 10.29578079426045571924076980334

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