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Properties

Label 1-5520-5520.2717-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.614 - 0.789i$
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  + (−0.909 − 0.415i)7-s + (0.281 − 0.959i)11-s + (0.415 + 0.909i)13-s + (0.540 − 0.841i)17-s + (−0.540 − 0.841i)19-s + (−0.540 + 0.841i)29-s + (−0.654 + 0.755i)31-s + (−0.142 + 0.989i)37-s + (−0.142 − 0.989i)41-s + (0.654 + 0.755i)43-s − i·47-s + (0.654 + 0.755i)49-s + (0.415 − 0.909i)53-s + (0.909 − 0.415i)59-s + (−0.755 − 0.654i)61-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)7-s + (0.281 − 0.959i)11-s + (0.415 + 0.909i)13-s + (0.540 − 0.841i)17-s + (−0.540 − 0.841i)19-s + (−0.540 + 0.841i)29-s + (−0.654 + 0.755i)31-s + (−0.142 + 0.989i)37-s + (−0.142 − 0.989i)41-s + (0.654 + 0.755i)43-s − i·47-s + (0.654 + 0.755i)49-s + (0.415 − 0.909i)53-s + (0.909 − 0.415i)59-s + (−0.755 − 0.654i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.614 - 0.789i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (2717, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.614 - 0.789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4202742008 - 0.8595880295i\)
\(L(\frac12)\) \(\approx\) \(0.4202742008 - 0.8595880295i\)
\(L(1)\) \(\approx\) \(0.8885869067 - 0.1756941190i\)
\(L(1)\) \(\approx\) \(0.8885869067 - 0.1756941190i\)

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 \( 1 + (-0.909 - 0.415i)T \)
11 \( 1 + (0.281 - 0.959i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.540 - 0.841i)T \)
19 \( 1 + (-0.540 - 0.841i)T \)
29 \( 1 + (-0.540 + 0.841i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (-0.142 + 0.989i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.415 - 0.909i)T \)
59 \( 1 + (0.909 - 0.415i)T \)
61 \( 1 + (-0.755 - 0.654i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.540 + 0.841i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.989 - 0.142i)T \)
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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

−18.1918090137574439781516432251, −17.34505431632353665316180679086, −16.842213415047652480229312104180, −16.12061443110636125790047428578, −15.342080892308140572847189138845, −14.97021231399341571053788265904, −14.26147023287999509839023439919, −13.23500661862570472728966628550, −12.76095597871359564748400399451, −12.33566541626462345431091788252, −11.53491952843845195776857374777, −10.54395846921080105381548791613, −10.14768992363871767111604176388, −9.39550970417222953127818332509, −8.790750314654903720886453286066, −7.83971763672317143578128908485, −7.41963742356461483801903753879, −6.282796073528804983041779982718, −5.97806392930653620852526479870, −5.227755028430700035697831838490, −4.00114271642382909770434335204, −3.76264723773783225696919506355, −2.68294546134661409526116030572, −2.01403074379866556142649196493, −1.02634742924055238927081866423, 0.2800186305566177477080411465, 1.18925023871403735177811989617, 2.18974505958207982141047303990, 3.23376873738198720352380328011, 3.55675094294503489506771845235, 4.49534500188355025910140299597, 5.33132003016096152262966313175, 6.104964003436345171930769482355, 6.882144771104652386613382665785, 7.15900896323463659694622488554, 8.35636053980268506390249565537, 8.942634615101840806531449160958, 9.488496177489939998738069582365, 10.288508115018879225690675334091, 11.036818045390151577579641889232, 11.53859859075806215953596550596, 12.36991849286165123520135731124, 13.08946162508795699859044484166, 13.70332108248690684811732328278, 14.178125769372697886856426243209, 14.9602890933462373769249976154, 15.98532177194247292779271220719, 16.23121402431134815426661176749, 16.83276513335760153186278965180, 17.5419924817337785867065281424

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