↑

Properties

Label 1-5520-5520.3227-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.988 + 0.149i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.540 − 0.841i)7-s + (0.989 + 0.142i)11-s + (−0.841 − 0.540i)13-s + (0.281 − 0.959i)17-s + (−0.281 − 0.959i)19-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (−0.654 − 0.755i)37-s + (−0.654 + 0.755i)41-s + (0.415 − 0.909i)43-s − i·47-s + (−0.415 + 0.909i)49-s + (0.841 − 0.540i)53-s + (0.540 − 0.841i)59-s + (−0.909 + 0.415i)61-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)7-s + (0.989 + 0.142i)11-s + (−0.841 − 0.540i)13-s + (0.281 − 0.959i)17-s + (−0.281 − 0.959i)19-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (−0.654 − 0.755i)37-s + (−0.654 + 0.755i)41-s + (0.415 − 0.909i)43-s − i·47-s + (−0.415 + 0.909i)49-s + (0.841 − 0.540i)53-s + (0.540 − 0.841i)59-s + (−0.909 + 0.415i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.988 + 0.149i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (3227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.988 + 0.149i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04447481436 - 0.5929804604i\)
\(L(\frac12)\) \(\approx\) \(-0.04447481436 - 0.5929804604i\)
\(L(1)\) \(\approx\) \(0.8398323917 - 0.2396043611i\)
\(L(1)\) \(\approx\) \(0.8398323917 - 0.2396043611i\)

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 \( 1 + (-0.540 - 0.841i)T \)
11 \( 1 + (0.989 + 0.142i)T \)
13 \( 1 + (-0.841 - 0.540i)T \)
17 \( 1 + (0.281 - 0.959i)T \)
19 \( 1 + (-0.281 - 0.959i)T \)
29 \( 1 + (-0.281 + 0.959i)T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + (-0.654 - 0.755i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.841 - 0.540i)T \)
59 \( 1 + (0.540 - 0.841i)T \)
61 \( 1 + (-0.909 + 0.415i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (-0.281 - 0.959i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.755 + 0.654i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.3201851318602574973844587628, −17.42267629878192056944125816864, −16.82028277338768985118353704132, −16.42694377591898305128297192862, −15.45213641888258379834778651433, −14.923594239510687183164622464411, −14.37020131547121674996280158829, −13.63447819804640649115055652061, −12.77046350695152173535328690434, −12.04384846317837596937381113517, −11.94171089871795371833452521705, −10.86244688205614748172968421210, −10.02463398214313514588163524532, −9.58329264097242601361939955094, −8.717374905008678919047808778277, −8.33346391527664604860722234335, −7.24261953603279454133046397696, −6.6383873596005264748762018604, −5.91852814925943505505333056182, −5.37781446124241773835949783405, −4.292166984687339319108431845293, −3.73519801624792067245801318401, −2.8903543742875413147919198868, −1.998950080683426856531343756488, −1.36899744191233753387741397400, 0.16402262789915453922525967730, 1.00062051714085272311171714695, 2.04167448533149754788296534205, 2.95719500278426135858424408308, 3.59945350210274911298299508382, 4.41846370942513702295991592278, 5.070563658589130991173278891060, 5.93733951839454167397325331050, 6.88552485556538081986166586032, 7.16458294655379849748855593566, 7.90032466194557699859692688444, 9.05738449561433114094623145893, 9.359903744941236414144984657726, 10.16939185421865236327349200669, 10.78110599974020531052352361173, 11.57477988155436229240172555521, 12.20307948228992606208623026990, 12.97338382816896605365087318356, 13.47675107767284164779080438061, 14.359405461841581148095069284877, 14.69647124507202374332799783422, 15.6447896834820871314601511926, 16.235054279295725418385611223196, 17.027793204316269006831238191711, 17.30020682865246728638367643040

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