Properties

Label 2-240-3.2-c2-0-13
Degree $2$
Conductor $240$
Sign $0.288 + 0.957i$
Analytic cond. $6.53952$
Root an. cond. $2.55724$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.87 − 0.864i)3-s − 2.23i·5-s − 9.02·7-s + (7.50 − 4.96i)9-s − 21.8i·11-s + 21.6·13-s + (−1.93 − 6.42i)15-s − 12.1i·17-s − 3.03·19-s + (−25.9 + 7.80i)21-s + 28.5i·23-s − 5.00·25-s + (17.2 − 20.7i)27-s + 12.0i·29-s − 2.19·31-s + ⋯
L(s)  = 1  + (0.957 − 0.288i)3-s − 0.447i·5-s − 1.28·7-s + (0.833 − 0.551i)9-s − 1.98i·11-s + 1.66·13-s + (−0.128 − 0.428i)15-s − 0.712i·17-s − 0.159·19-s + (−1.23 + 0.371i)21-s + 1.24i·23-s − 0.200·25-s + (0.639 − 0.768i)27-s + 0.415i·29-s − 0.0706·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.288 + 0.957i$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ 0.288 + 0.957i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.54412 - 1.14786i\)
\(L(\frac12)\) \(\approx\) \(1.54412 - 1.14786i\)
\(L(2)\) 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 \( 1 + (-2.87 + 0.864i)T \)
5 \( 1 + 2.23iT \)
good7 \( 1 + 9.02T + 49T^{2} \)
11 \( 1 + 21.8iT - 121T^{2} \)
13 \( 1 - 21.6T + 169T^{2} \)
17 \( 1 + 12.1iT - 289T^{2} \)
19 \( 1 + 3.03T + 361T^{2} \)
23 \( 1 - 28.5iT - 529T^{2} \)
29 \( 1 - 12.0iT - 841T^{2} \)
31 \( 1 + 2.19T + 961T^{2} \)
37 \( 1 - 0.839T + 1.36e3T^{2} \)
41 \( 1 - 35.5iT - 1.68e3T^{2} \)
43 \( 1 - 12.7T + 1.84e3T^{2} \)
47 \( 1 + 22.5iT - 2.20e3T^{2} \)
53 \( 1 - 9.13iT - 2.80e3T^{2} \)
59 \( 1 - 80.4iT - 3.48e3T^{2} \)
61 \( 1 + 57.8T + 3.72e3T^{2} \)
67 \( 1 - 63.0T + 4.48e3T^{2} \)
71 \( 1 + 17.0iT - 5.04e3T^{2} \)
73 \( 1 - 52.1T + 5.32e3T^{2} \)
79 \( 1 - 7.46T + 6.24e3T^{2} \)
83 \( 1 - 82.3iT - 6.88e3T^{2} \)
89 \( 1 + 27.5iT - 7.92e3T^{2} \)
97 \( 1 - 114.T + 9.40e3T^{2} \)
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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

−11.77299933336666510238472314941, −10.72745983103159802168656860500, −9.416602261890153393730132560512, −8.846784724562201043348264790700, −7.983125746849326870382871954063, −6.60157516447708268029764743004, −5.76321181927646337159782677508, −3.72232248968304129089854029332, −3.09979066058086892557914610295, −0.990720479328746970168056589680, 2.08989193463306651220665539794, 3.43847213038181655221817954953, 4.37591476684258090906478928633, 6.26228301921994693204046121100, 7.07758090338624083871534399116, 8.268767024400242557574439032612, 9.308765080582303248357760078724, 10.06178018157075165745074702609, 10.79976768984194059137548397291, 12.49152330502681098327817877606

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