Properties

Label 2-240-5.3-c2-0-0
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 + (−3.44 + 3.44i)7-s + 2.99i·9-s − 11.3·11-s + (−5.55 − 5.55i)13-s + (−3.55 − 7.89i)15-s + (−17.3 + 17.3i)17-s + 8.69i·19-s − 8.44·21-s + (−11.5 − 11.5i)23-s + (18.6 + 16.5i)25-s + (−3.67 + 3.67i)27-s − 35.1i·29-s − 10.6·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.934 − 0.355i)5-s + (−0.492 + 0.492i)7-s + 0.333i·9-s − 1.03·11-s + (−0.426 − 0.426i)13-s + (−0.236 − 0.526i)15-s + (−1.02 + 1.02i)17-s + 0.457i·19-s − 0.402·21-s + (−0.502 − 0.502i)23-s + (0.747 + 0.663i)25-s + (−0.136 + 0.136i)27-s − 1.21i·29-s − 0.345·31-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

\(L(\frac{3}{2})\) \(\approx\) \(0.0205645 + 0.313166i\)
\(L(\frac12)\) \(\approx\) \(0.0205645 + 0.313166i\)
\(L(2)\) 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 \( 1 + (3.44 - 3.44i)T - 49iT^{2} \)
11 \( 1 + 11.3T + 121T^{2} \)
13 \( 1 + (5.55 + 5.55i)T + 169iT^{2} \)
17 \( 1 + (17.3 - 17.3i)T - 289iT^{2} \)
19 \( 1 - 8.69iT - 361T^{2} \)
23 \( 1 + (11.5 + 11.5i)T + 529iT^{2} \)
29 \( 1 + 35.1iT - 841T^{2} \)
31 \( 1 + 10.6T + 961T^{2} \)
37 \( 1 + (6.04 - 6.04i)T - 1.36e3iT^{2} \)
41 \( 1 - 0.696T + 1.68e3T^{2} \)
43 \( 1 + (-26.4 - 26.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (44.2 - 44.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (0.696 + 0.696i)T + 2.80e3iT^{2} \)
59 \( 1 + 39.9iT - 3.48e3T^{2} \)
61 \( 1 - 5.90T + 3.72e3T^{2} \)
67 \( 1 + (-45.1 + 45.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (-77.7 - 77.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (13.1 + 13.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 82.1iT - 7.92e3T^{2} \)
97 \( 1 + (24.5 - 24.5i)T - 9.40e3iT^{2} \)
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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

−12.58989633990920535086257766923, −11.34379217634036986586075057725, −10.43265760198953227742029429913, −9.429395991553941849866521795435, −8.323685159176518791250024398131, −7.78046511787214570316113629392, −6.24157634533574064888925930521, −4.92694710288158131367275526607, −3.83087607854624593658669646398, −2.51138712799957098980902801806, 0.14765568633595761737213549879, 2.51665413217992251815533356522, 3.71164184470826098600082485207, 5.03254301052183749012983187564, 6.82705523775956883503006196835, 7.29451298350207619045197711856, 8.368092775420295405235965785671, 9.445831724325535723128165655883, 10.59362811138997890927589748131, 11.45587346128434502709280498683

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