Properties

Label 2-240-60.59-c1-0-0
Degree $2$
Conductor $240$
Sign $-0.558 - 0.829i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
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  + (−1.41 − i)3-s + (−1.73 + 1.41i)5-s + (1.00 + 2.82i)9-s − 4.89·11-s + 4.89i·13-s + (3.86 − 0.267i)15-s − 3.46·17-s + 3.46i·19-s + 6i·23-s + (0.999 − 4.89i)25-s + (1.41 − 5.00i)27-s − 2.82i·29-s − 3.46i·31-s + (6.92 + 4.89i)33-s − 4.89i·37-s + ⋯
L(s)  = 1  + (−0.816 − 0.577i)3-s + (−0.774 + 0.632i)5-s + (0.333 + 0.942i)9-s − 1.47·11-s + 1.35i·13-s + (0.997 − 0.0691i)15-s − 0.840·17-s + 0.794i·19-s + 1.25i·23-s + (0.199 − 0.979i)25-s + (0.272 − 0.962i)27-s − 0.525i·29-s − 0.622i·31-s + (1.20 + 0.852i)33-s − 0.805i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.558 - 0.829i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.558 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169004 + 0.317631i\)
\(L(\frac12)\) \(\approx\) \(0.169004 + 0.317631i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + i)T \)
5 \( 1 + (1.73 - 1.41i)T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 - 4.89iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 4.89iT - 37T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 8.48T + 67T^{2} \)
71 \( 1 - 9.79T + 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.29842972763067572273573034576, −11.42134135491692517718233535634, −10.89201704853482193869461871279, −9.796055159475602554161942082646, −8.206512529521974946875474537125, −7.41244561583080313598238892149, −6.54147319896468511900706437894, −5.33574215362576482465350029422, −4.04350852873106186643099601937, −2.23382302473330799143475228313, 0.30577859808705926005050869219, 3.14342414870859641184431479662, 4.70939093318512291498847350140, 5.23170425648756965449813761998, 6.67697981791199722123826300309, 7.958268692401170129599735329895, 8.806048196688570406636679769706, 10.19344748194443834927611484899, 10.78462466243120143156101306806, 11.72595960861378946355439975508

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