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Properties

Label 2-605-55.42-c0-0-0
Degree $2$
Conductor $605$
Sign $-0.995 - 0.0965i$
Analytic cond. $0.301934$
Root an. cond. $0.549485$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.221 + 1.39i)3-s + (−0.587 + 0.809i)4-s + (−0.951 + 0.309i)5-s + (−0.951 − 0.309i)9-s + (−1 − i)12-s + (−0.221 − 1.39i)15-s + (−0.309 − 0.951i)16-s + (0.309 − 0.951i)20-s + (−1 + i)23-s + (0.809 − 0.587i)25-s + (0.809 − 0.587i)36-s + (0.221 + 1.39i)37-s + 45-s + (−1.39 − 0.221i)47-s + (1.39 − 0.221i)48-s + (0.951 − 0.309i)49-s + ⋯
L(s)  = 1  + (−0.221 + 1.39i)3-s + (−0.587 + 0.809i)4-s + (−0.951 + 0.309i)5-s + (−0.951 − 0.309i)9-s + (−1 − i)12-s + (−0.221 − 1.39i)15-s + (−0.309 − 0.951i)16-s + (0.309 − 0.951i)20-s + (−1 + i)23-s + (0.809 − 0.587i)25-s + (0.809 − 0.587i)36-s + (0.221 + 1.39i)37-s + 45-s + (−1.39 − 0.221i)47-s + (1.39 − 0.221i)48-s + (0.951 − 0.309i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.995 - 0.0965i$
Analytic conductor: \(0.301934\)
Root analytic conductor: \(0.549485\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (372, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :0),\ -0.995 - 0.0965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5136037914\)
\(L(\frac12)\) \(\approx\) \(0.5136037914\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 \)
good2 \( 1 + (0.587 - 0.809i)T^{2} \)
3 \( 1 + (0.221 - 1.39i)T + (-0.951 - 0.309i)T^{2} \)
7 \( 1 + (-0.951 + 0.309i)T^{2} \)
13 \( 1 + (-0.587 + 0.809i)T^{2} \)
17 \( 1 + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (1 - i)T - iT^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.221 - 1.39i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (-0.642 - 1.26i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-1 - i)T + iT^{2} \)
71 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + 2iT - T^{2} \)
97 \( 1 + (1.26 - 0.642i)T + (0.587 - 0.809i)T^{2} \)
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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

−11.39891696530327906551868542134, −10.28896866742968965434858274336, −9.661847341795667530232545571551, −8.680846807514395689489743206579, −7.972002069276462539150117508546, −6.99779266629895196945993201176, −5.50344381153514975524909395519, −4.48087983787020986270055012372, −3.89255125114819955002376219707, −3.05004765402552925916788092080, 0.63173197566369554946515770389, 2.04423283722651196890436394849, 3.87965630933932625036981791227, 4.95825131402453036523765982036, 6.03834768574954998461258949550, 6.83369878006609216841574686530, 7.83568195764347856991932759388, 8.461398860272049482608893734867, 9.475041354262135214182481635311, 10.59647477724589526415723743851

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