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Properties

Label 1-5520-5520.2699-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.348 + 0.937i$
Analytic cond. $25.6347$
Root an. cond. $25.6347$
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  + (−0.415 − 0.909i)7-s + (−0.281 − 0.959i)11-s + (0.909 + 0.415i)13-s + (0.841 − 0.540i)17-s + (−0.540 + 0.841i)19-s + (0.540 + 0.841i)29-s + (0.654 + 0.755i)31-s + (−0.989 + 0.142i)37-s + (−0.142 + 0.989i)41-s + (−0.755 − 0.654i)43-s − 47-s + (−0.654 + 0.755i)49-s + (−0.909 + 0.415i)53-s + (0.909 + 0.415i)59-s + (−0.755 + 0.654i)61-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)7-s + (−0.281 − 0.959i)11-s + (0.909 + 0.415i)13-s + (0.841 − 0.540i)17-s + (−0.540 + 0.841i)19-s + (0.540 + 0.841i)29-s + (0.654 + 0.755i)31-s + (−0.989 + 0.142i)37-s + (−0.142 + 0.989i)41-s + (−0.755 − 0.654i)43-s − 47-s + (−0.654 + 0.755i)49-s + (−0.909 + 0.415i)53-s + (0.909 + 0.415i)59-s + (−0.755 + 0.654i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.348 + 0.937i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (2699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.348 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9350882184 + 0.6499895422i\)
\(L(\frac12)\) \(\approx\) \(0.9350882184 + 0.6499895422i\)
\(L(1)\) \(\approx\) \(0.9660412107 + 0.01707572609i\)
\(L(1)\) \(\approx\) \(0.9660412107 + 0.01707572609i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (-0.281 - 0.959i)T \)
13 \( 1 + (0.909 + 0.415i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (-0.540 + 0.841i)T \)
29 \( 1 + (0.540 + 0.841i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + (-0.989 + 0.142i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + (0.909 + 0.415i)T \)
61 \( 1 + (-0.755 + 0.654i)T \)
67 \( 1 + (-0.281 + 0.959i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (-0.415 + 0.909i)T \)
83 \( 1 + (0.989 - 0.142i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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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

−17.69499606752439883335053004352, −17.2449302830085393980717974635, −16.31749147583384149640191375647, −15.56639287281772839945319608687, −15.30144848810962592555262366926, −14.602054574438883135021938223719, −13.644841462716527888393441079782, −13.046863544546888563118457526874, −12.44732696722493753342812627977, −11.88605810460852150470572936302, −11.07206395039684186509495159666, −10.307462417776888324693828308478, −9.71620960529784037239916368534, −9.03028034146380576052022335647, −8.219350756135540576394331198820, −7.79591559365559100834183563823, −6.58504031976573219815383007040, −6.3165798447989643076666939651, −5.35149060074652073311496835684, −4.79930441311586931919081671909, −3.79831096017656350444046475495, −3.10805972048764519785201789859, −2.27663652881435555459593819133, −1.59622291528873294522071341741, −0.321255482685520826495493001995, 0.991042503989259897023802884210, 1.51225094954186524844361287316, 2.88218035856592116886814228417, 3.40602822385733986746490269819, 4.04466135232359585446606615538, 4.98149726059987971585826727436, 5.72226003129427170187174511876, 6.56244219877958691814083270560, 6.94933987923163356467436931302, 8.08858906213866063385992644987, 8.36373041404160643626223907886, 9.296003149677475115466504969829, 10.143131511391686178603766063013, 10.55436287725990831430840106268, 11.275971714815124143810062522057, 12.01024513725431471794369938209, 12.7628977662748699773263648285, 13.50038503155217776161536787518, 13.985253129504867643278636199827, 14.47301727188918929973208682563, 15.52643826763395686866907287064, 16.269743431366398589117659342647, 16.47374107922445748946643807884, 17.202899671878910094006345283382, 18.11180957940994360212649712201

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