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Properties

Label 1-5520-5520.4109-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.691 + 0.722i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (4109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.691 + 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9295499692 + 0.3970637847i\)
\(L(\frac12)\) \(\approx\) \(0.9295499692 + 0.3970637847i\)
\(L(1)\) \(\approx\) \(0.8777689217 + 0.02715852242i\)
\(L(1)\) \(\approx\) \(0.8777689217 + 0.02715852242i\)

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

−17.77092669074060076629835713391, −17.1352584923317417568610936745, −16.28203872112050211393762136049, −15.95121298998174387837685153155, −15.22693196120527197455562173115, −14.240974236769247001860927878891, −14.11087559818173746323377505168, −13.05424100218613937776776674042, −12.39561315950645520815048646287, −11.888718725900788720587934819341, −11.19403795520691618001414851690, −10.44548461534557093884611644074, −9.62343467950912362662465684497, −8.89640904941098212858018375241, −8.684798122930414246325903029984, −7.48261162566918504937085027652, −6.97082377387615832576648157731, −6.11354224833967810519749512154, −5.46382936860850085810410970321, −4.887181465333693885316349055892, −3.76288558274457399556606014550, −3.21741566583957251944879812150, −2.32609415580179552974293213217, −1.66086863887853317966492275160, −0.34501816074734132988723237138, 0.709606909895875936658635229737, 1.77590981382089143487348011699, 2.53856278623422256432313730279, 3.39435303173162940811038072172, 4.32310969348590941265755347761, 4.64096758521584278011375513713, 5.68911110061380714790493395823, 6.47188634972037002835711696994, 7.32912479567043365213838307782, 7.45688385559355893073560950659, 8.5388349410826924200600931994, 9.52483281268386766814422572854, 9.73155217096991369387345841187, 10.62611410586160751199563409438, 11.20400087667040932751218145350, 12.03866315833611538612199866953, 12.7718644168639994785428360493, 13.236079094320100517649073624292, 13.95064162997260955908217681634, 14.73648161132965717350427905140, 15.310095423128467190372391960346, 15.88331064684507586186027999426, 16.88462492746290090201725749058, 17.241585799031088239320729617034, 17.75547789504932982966973092141

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