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Properties

Label 1-2404-2404.1123-r0-0-0
Degree $1$
Conductor $2404$
Sign $-0.705 - 0.708i$
Analytic cond. $11.1641$
Root an. cond. $11.1641$
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  + (−0.968 − 0.248i)3-s − i·5-s + (0.750 − 0.661i)7-s + (0.876 + 0.481i)9-s + (0.940 − 0.338i)11-s + (0.951 − 0.309i)13-s + (−0.248 + 0.968i)15-s + (−0.156 − 0.987i)17-s + (−0.218 + 0.975i)19-s + (−0.891 + 0.453i)21-s + (0.125 − 0.992i)23-s − 25-s + (−0.728 − 0.684i)27-s + (−0.975 − 0.218i)29-s + (0.790 − 0.612i)31-s + ⋯
L(s)  = 1  + (−0.968 − 0.248i)3-s − i·5-s + (0.750 − 0.661i)7-s + (0.876 + 0.481i)9-s + (0.940 − 0.338i)11-s + (0.951 − 0.309i)13-s + (−0.248 + 0.968i)15-s + (−0.156 − 0.987i)17-s + (−0.218 + 0.975i)19-s + (−0.891 + 0.453i)21-s + (0.125 − 0.992i)23-s − 25-s + (−0.728 − 0.684i)27-s + (−0.975 − 0.218i)29-s + (0.790 − 0.612i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2404 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2404 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2404\)    =    \(2^{2} \cdot 601\)
Sign: $-0.705 - 0.708i$
Analytic conductor: \(11.1641\)
Root analytic conductor: \(11.1641\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2404} (1123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2404,\ (0:\ ),\ -0.705 - 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5164799791 - 1.244071733i\)
\(L(\frac12)\) \(\approx\) \(0.5164799791 - 1.244071733i\)
\(L(1)\) \(\approx\) \(0.8389499555 - 0.4507706965i\)
\(L(1)\) \(\approx\) \(0.8389499555 - 0.4507706965i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
601 \( 1 \)
good3 \( 1 + (-0.968 - 0.248i)T \)
5 \( 1 - iT \)
7 \( 1 + (0.750 - 0.661i)T \)
11 \( 1 + (0.940 - 0.338i)T \)
13 \( 1 + (0.951 - 0.309i)T \)
17 \( 1 + (-0.156 - 0.987i)T \)
19 \( 1 + (-0.218 + 0.975i)T \)
23 \( 1 + (0.125 - 0.992i)T \)
29 \( 1 + (-0.975 - 0.218i)T \)
31 \( 1 + (0.790 - 0.612i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.860 - 0.509i)T \)
43 \( 1 + (0.278 - 0.960i)T \)
47 \( 1 + (-0.728 - 0.684i)T \)
53 \( 1 + (-0.960 + 0.278i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.998 + 0.0627i)T \)
67 \( 1 + (-0.0627 - 0.998i)T \)
71 \( 1 + (-0.987 - 0.156i)T \)
73 \( 1 + (-0.0941 + 0.995i)T \)
79 \( 1 + (0.917 - 0.397i)T \)
83 \( 1 + (0.999 + 0.0314i)T \)
89 \( 1 + (-0.728 + 0.684i)T \)
97 \( 1 + (0.891 - 0.453i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.488676459643917438866833287930, −19.112974970748738962623192741843, −18.10546499215552947441953463971, −17.682413105198785823310437413371, −17.26716828563350626501549561342, −16.11588061330805385420771140860, −15.53290086554559265892233346343, −14.84123294599257783383748176584, −14.31423912508213352214657079363, −13.23453530061481287533860023667, −12.45485878211685296419265192537, −11.44913877139179172713867694293, −11.27989410603906470857428313520, −10.64348029855734874640218433233, −9.58084264633584516269711721617, −9.01172069992703300061033224807, −7.923974044275017980408869878006, −7.04089750710495864455939597988, −6.29484469982535562369227429046, −5.86344343006147793482101833644, −4.801972234422271416806173293698, −4.058589275165943359172567859517, −3.25112405058715376878438091398, −1.95306394687285639413333721665, −1.31040960296667803179219582386, 0.57866322472431373252043787644, 1.20570720554661622900271871057, 2.002491814298004349581115226077, 3.67872098394529680930181959938, 4.32760527610093579763328582912, 5.00818420387960825257542684661, 5.85029272336757931550895424391, 6.48676312374343050184093693326, 7.4752457721950446255789857344, 8.181206860737447015770567507573, 8.92731258512924982348343042981, 9.88269970714372453019452397513, 10.691456246121931637414095856543, 11.42193219765408434192665703649, 11.92281100786308662897738860688, 12.689133547079845494794764756547, 13.52232482883405804951830677075, 13.97520252459842778107557124602, 15.064858948681887074215477171316, 16.00212348453101565735476923951, 16.61513706850703999813980549307, 17.029522011714586563106675083830, 17.68954528467091993248929599138, 18.4932116322876727320144150111, 19.11040378129983383066998836994

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