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Properties

Label 2-2520-7.4-c1-0-15
Degree $2$
Conductor $2520$
Sign $0.991 - 0.126i$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (−2 + 1.73i)7-s + (1.5 − 2.59i)11-s + 13-s + (3.5 + 6.06i)19-s + (−2.5 − 4.33i)23-s + (−0.499 + 0.866i)25-s + (−3 + 5.19i)31-s + (2.5 + 0.866i)35-s + (−1.5 − 2.59i)37-s + 3·41-s + 8·43-s + (−0.5 − 0.866i)47-s + (1.00 − 6.92i)49-s + (2.5 − 4.33i)53-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.755 + 0.654i)7-s + (0.452 − 0.783i)11-s + 0.277·13-s + (0.802 + 1.39i)19-s + (−0.521 − 0.902i)23-s + (−0.0999 + 0.173i)25-s + (−0.538 + 0.933i)31-s + (0.422 + 0.146i)35-s + (−0.246 − 0.427i)37-s + 0.468·41-s + 1.21·43-s + (−0.0729 − 0.126i)47-s + (0.142 − 0.989i)49-s + (0.343 − 0.594i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.532386769\)
\(L(\frac12)\) \(\approx\) \(1.532386769\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2 - 1.73i)T \)
good11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.5 + 4.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.5 + 4.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.902864024088460803182879394994, −8.301940240250920807783633374213, −7.47426403077949028117787254963, −6.43816277742489665642610738531, −5.88581021447571321201622606447, −5.14080045295102106053661375082, −3.89184152053049030664057411143, −3.36831930887904137204450785417, −2.17116471879093239034055719207, −0.831904075491864375468347720537, 0.74055757516995056951015704646, 2.17761410624789828117848272655, 3.28250322732277928044848727034, 3.96807218411655335046076459677, 4.85570130901002677467830035324, 5.93250451670301538326087697393, 6.71230022867760360007390635906, 7.33266499804856655973525924971, 7.891368604562447538540632708795, 9.260265202689458243795270816822

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