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Properties

Label 2-240-5.3-c2-0-7
Degree $2$
Conductor $240$
Sign $0.991 + 0.130i$
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  + (−1.22 − 1.22i)3-s + (4.67 + 1.77i)5-s + (0.550 − 0.550i)7-s + 2.99i·9-s + 1.55·11-s + (9.55 + 9.55i)13-s + (−3.55 − 7.89i)15-s + (11.1 − 11.1i)17-s − 12.6i·19-s − 1.34·21-s + (21.3 + 21.3i)23-s + (18.6 + 16.5i)25-s + (3.67 − 3.67i)27-s − 44.0i·29-s + 44.4·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.934 + 0.355i)5-s + (0.0786 − 0.0786i)7-s + 0.333i·9-s + 0.140·11-s + (0.734 + 0.734i)13-s + (−0.236 − 0.526i)15-s + (0.655 − 0.655i)17-s − 0.668i·19-s − 0.0642·21-s + (0.928 + 0.928i)23-s + (0.747 + 0.663i)25-s + (0.136 − 0.136i)27-s − 1.51i·29-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ 0.991 + 0.130i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.69575 - 0.111354i\)
\(L(\frac12)\) \(\approx\) \(1.69575 - 0.111354i\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (-4.67 - 1.77i)T \)
good7 \( 1 + (-0.550 + 0.550i)T - 49iT^{2} \)
11 \( 1 - 1.55T + 121T^{2} \)
13 \( 1 + (-9.55 - 9.55i)T + 169iT^{2} \)
17 \( 1 + (-11.1 + 11.1i)T - 289iT^{2} \)
19 \( 1 + 12.6iT - 361T^{2} \)
23 \( 1 + (-21.3 - 21.3i)T + 529iT^{2} \)
29 \( 1 + 44.0iT - 841T^{2} \)
31 \( 1 - 44.4T + 961T^{2} \)
37 \( 1 + (-20.6 + 20.6i)T - 1.36e3iT^{2} \)
41 \( 1 + 48.2T + 1.68e3T^{2} \)
43 \( 1 + (-36.2 - 36.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (42.5 - 42.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (54.4 + 54.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 47.4iT - 3.48e3T^{2} \)
61 \( 1 + 59.8T + 3.72e3T^{2} \)
67 \( 1 + (81.2 - 81.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 87.5T + 5.04e3T^{2} \)
73 \( 1 + (-75.9 - 75.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 97.3iT - 6.24e3T^{2} \)
83 \( 1 + (41.0 + 41.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 52.2iT - 7.92e3T^{2} \)
97 \( 1 + (37 - 37i)T - 9.40e3iT^{2} \)
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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

−11.67553634608602922012666876827, −11.12451470141217171000237289492, −9.909634329174639754858461434012, −9.158717830000846545173896628751, −7.77488917199181393218541443340, −6.67769851114011672269424652429, −5.91856876326702440171120733018, −4.68233255335959513206543665732, −2.88075086577379548882858167838, −1.32844154521462382965347329959, 1.29105787961853537195263083430, 3.20072044383043660764411605927, 4.75821520727822079644926806783, 5.72777998598385733828651653845, 6.58391123664760406837101209374, 8.215755654473045014079620963926, 9.043402422462823878162390797654, 10.24414647075233640779668857949, 10.64052757539304225904853281879, 12.02678972256527805745553936932

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