Properties

Label 2-240-80.29-c1-0-13
Degree $2$
Conductor $240$
Sign $-0.227 + 0.973i$
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  + (−0.750 − 1.19i)2-s + (−0.707 + 0.707i)3-s + (−0.874 + 1.79i)4-s + (1.07 − 1.95i)5-s + (1.37 + 0.317i)6-s − 1.22·7-s + (2.81 − 0.302i)8-s − 1.00i·9-s + (−3.15 + 0.178i)10-s + (1.38 − 1.38i)11-s + (−0.654 − 1.89i)12-s + (2.12 − 2.12i)13-s + (0.916 + 1.46i)14-s + (0.623 + 2.14i)15-s + (−2.47 − 3.14i)16-s − 6.00i·17-s + ⋯
L(s)  = 1  + (−0.530 − 0.847i)2-s + (−0.408 + 0.408i)3-s + (−0.437 + 0.899i)4-s + (0.481 − 0.876i)5-s + (0.562 + 0.129i)6-s − 0.461·7-s + (0.994 − 0.106i)8-s − 0.333i·9-s + (−0.998 + 0.0565i)10-s + (0.416 − 0.416i)11-s + (−0.188 − 0.545i)12-s + (0.588 − 0.588i)13-s + (0.244 + 0.391i)14-s + (0.161 + 0.554i)15-s + (−0.618 − 0.786i)16-s − 1.45i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.227 + 0.973i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.227 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.494402 - 0.622986i\)
\(L(\frac12)\) \(\approx\) \(0.494402 - 0.622986i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.750 + 1.19i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.07 + 1.95i)T \)
good7 \( 1 + 1.22T + 7T^{2} \)
11 \( 1 + (-1.38 + 1.38i)T - 11iT^{2} \)
13 \( 1 + (-2.12 + 2.12i)T - 13iT^{2} \)
17 \( 1 + 6.00iT - 17T^{2} \)
19 \( 1 + (3.06 + 3.06i)T + 19iT^{2} \)
23 \( 1 - 2.90T + 23T^{2} \)
29 \( 1 + (-3.18 - 3.18i)T + 29iT^{2} \)
31 \( 1 + 3.88T + 31T^{2} \)
37 \( 1 + (-2.44 - 2.44i)T + 37iT^{2} \)
41 \( 1 + 2.38iT - 41T^{2} \)
43 \( 1 + (-9.00 - 9.00i)T + 43iT^{2} \)
47 \( 1 + 0.586iT - 47T^{2} \)
53 \( 1 + (2.36 + 2.36i)T + 53iT^{2} \)
59 \( 1 + (-8.43 + 8.43i)T - 59iT^{2} \)
61 \( 1 + (-9.98 - 9.98i)T + 61iT^{2} \)
67 \( 1 + (3.82 - 3.82i)T - 67iT^{2} \)
71 \( 1 - 11.5iT - 71T^{2} \)
73 \( 1 - 1.31T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + (2.91 - 2.91i)T - 83iT^{2} \)
89 \( 1 + 9.58iT - 89T^{2} \)
97 \( 1 - 9.45iT - 97T^{2} \)
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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

−11.65563865328960228983654612657, −10.94042883475728588895253290375, −9.850704328408175581080899072159, −9.170342999966755209124026818838, −8.397532805968769230872711803165, −6.85438306526993844799951769115, −5.42978436492462153882613872718, −4.32279771300566024875871651815, −2.89120050012833134741477467682, −0.865096743341993992647880739877, 1.82615547323473480906988739677, 4.04929224825462626145114113248, 5.80874842624689226435455719634, 6.39702738360576554895217837104, 7.19437639530769826113861904896, 8.393722485981994868473579765092, 9.487278649906544689897875933149, 10.42836387488373034508990707379, 11.11178501201144898058097032896, 12.55186031873485233147402503066

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