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Properties

Label 2-605-55.4-c1-0-20
Degree $2$
Conductor $605$
Sign $0.273 - 0.961i$
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  + (3.15 + 1.02i)3-s + (0.618 + 1.90i)4-s + (−2.18 − 0.459i)5-s + (6.47 + 4.70i)9-s + 6.63i·12-s + (−6.43 − 3.69i)15-s + (−3.23 + 2.35i)16-s + (−0.477 − 4.44i)20-s − 3.31i·23-s + (4.57 + 2.01i)25-s + (9.74 + 13.4i)27-s + (−4.04 − 2.93i)31-s + (−4.94 + 15.2i)36-s + (9.46 − 3.07i)37-s + (−12 − 13.2i)45-s + ⋯
L(s)  = 1  + (1.82 + 0.591i)3-s + (0.309 + 0.951i)4-s + (−0.978 − 0.205i)5-s + (2.15 + 1.56i)9-s + 1.91i·12-s + (−1.66 − 0.953i)15-s + (−0.809 + 0.587i)16-s + (−0.106 − 0.994i)20-s − 0.691i·23-s + (0.915 + 0.402i)25-s + (1.87 + 2.58i)27-s + (−0.726 − 0.527i)31-s + (−0.824 + 2.53i)36-s + (1.55 − 0.505i)37-s + (−1.78 − 1.97i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.273 - 0.961i$
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: \((2,\ 605,\ (\ :1/2),\ 0.273 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96416 + 1.48404i\)
\(L(\frac12)\) \(\approx\) \(1.96416 + 1.48404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.18 + 0.459i)T \)
11 \( 1 \)
good2 \( 1 + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-3.15 - 1.02i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 3.31iT - 23T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.04 + 2.93i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-9.46 + 3.07i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (6.30 + 2.04i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.79 + 10.7i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.63 + 14.2i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.94iT - 67T^{2} \)
71 \( 1 + (-2.42 + 1.76i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (5.84 - 8.04i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.81278758149184138233052392383, −9.667164361955451713502835504129, −8.883483407451935012664895824137, −8.144523691513245357242210131038, −7.73995643743466524847673194815, −6.82681140241026514138555912836, −4.71820937235048679006014466127, −3.92264737607168538797286251096, −3.24053637709502913762680422962, −2.22543299750200485659716120009, 1.28244606398199249642868335926, 2.55058556294639325246723229561, 3.50886403207379151971609900768, 4.61156897956159086415183557603, 6.24876682739677906370930153475, 7.21430529987924752294633517786, 7.74024162802330033424038064088, 8.708184278589467341633024109703, 9.427539934332469875638756096393, 10.29431431586599256477721472237

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