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Properties

Label 2-240-15.14-c8-0-5
Degree $2$
Conductor $240$
Sign $0.235 + 0.971i$
Analytic cond. $97.7708$
Root an. cond. $9.88791$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (32.2 + 74.2i)3-s + (−615. − 107. i)5-s + 4.01e3i·7-s + (−4.47e3 + 4.79e3i)9-s + 1.13e4i·11-s − 1.46e4i·13-s + (−1.19e4 − 4.92e4i)15-s − 1.21e5·17-s − 2.28e4·19-s + (−2.98e5 + 1.29e5i)21-s − 4.52e5·23-s + (3.67e5 + 1.32e5i)25-s + (−5.00e5 − 1.77e5i)27-s + 9.25e5i·29-s + 1.24e6·31-s + ⋯
L(s)  = 1  + (0.398 + 0.917i)3-s + (−0.985 − 0.171i)5-s + 1.67i·7-s + (−0.682 + 0.730i)9-s + 0.774i·11-s − 0.513i·13-s + (−0.235 − 0.971i)15-s − 1.45·17-s − 0.175·19-s + (−1.53 + 0.666i)21-s − 1.61·23-s + (0.941 + 0.338i)25-s + (−0.942 − 0.334i)27-s + 1.30i·29-s + 1.34·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.235 + 0.971i$
Analytic conductor: \(97.7708\)
Root analytic conductor: \(9.88791\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :4),\ 0.235 + 0.971i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.4175291147\)
\(L(\frac12)\) \(\approx\) \(0.4175291147\)
\(L(5)\) 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 \( 1 + (-32.2 - 74.2i)T \)
5 \( 1 + (615. + 107. i)T \)
good7 \( 1 - 4.01e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.13e4iT - 2.14e8T^{2} \)
13 \( 1 + 1.46e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.21e5T + 6.97e9T^{2} \)
19 \( 1 + 2.28e4T + 1.69e10T^{2} \)
23 \( 1 + 4.52e5T + 7.83e10T^{2} \)
29 \( 1 - 9.25e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.24e6T + 8.52e11T^{2} \)
37 \( 1 - 9.96e3iT - 3.51e12T^{2} \)
41 \( 1 - 3.24e6iT - 7.98e12T^{2} \)
43 \( 1 - 9.86e5iT - 1.16e13T^{2} \)
47 \( 1 + 7.49e5T + 2.38e13T^{2} \)
53 \( 1 + 3.98e6T + 6.22e13T^{2} \)
59 \( 1 - 1.17e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.04e7T + 1.91e14T^{2} \)
67 \( 1 + 1.35e7iT - 4.06e14T^{2} \)
71 \( 1 - 1.35e7iT - 6.45e14T^{2} \)
73 \( 1 + 5.39e7iT - 8.06e14T^{2} \)
79 \( 1 + 1.08e7T + 1.51e15T^{2} \)
83 \( 1 - 5.54e7T + 2.25e15T^{2} \)
89 \( 1 + 8.17e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.22e8iT - 7.83e15T^{2} \)
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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

−11.52013765165793735361864789422, −10.44231207094212399521821732085, −9.357902304992026105256361600979, −8.596359306842205888471569339147, −7.926682437399256112394840092550, −6.35271734437789510742660934265, −5.08362246277009076085677258680, −4.32851950013530364110292839184, −3.05080998876375959996307574933, −2.09492802806145913248178311803, 0.11313602231763533707081561710, 0.78040946071286287631941901053, 2.25063009852967245917926993950, 3.68443643414917361371155777647, 4.30762908628182289306940760159, 6.32381048961898635897074256019, 7.00426124633131950981146209650, 7.923319021228308046725709559811, 8.560478520904887554450557118072, 10.01836302916097859912649659699

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