Properties

Label 2-240-16.13-c1-0-5
Degree $2$
Conductor $240$
Sign $-0.633 - 0.773i$
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  + (0.635 + 1.26i)2-s + (0.707 + 0.707i)3-s + (−1.19 + 1.60i)4-s + (−0.707 + 0.707i)5-s + (−0.443 + 1.34i)6-s + 1.41i·7-s + (−2.78 − 0.484i)8-s + 1.00i·9-s + (−1.34 − 0.443i)10-s + (1.11 − 1.11i)11-s + (−1.97 + 0.292i)12-s + (−0.271 − 0.271i)13-s + (−1.78 + 0.898i)14-s − 1.00·15-s + (−1.15 − 3.82i)16-s + 0.744·17-s + ⋯
L(s)  = 1  + (0.449 + 0.893i)2-s + (0.408 + 0.408i)3-s + (−0.595 + 0.803i)4-s + (−0.316 + 0.316i)5-s + (−0.181 + 0.548i)6-s + 0.534i·7-s + (−0.985 − 0.171i)8-s + 0.333i·9-s + (−0.424 − 0.140i)10-s + (0.335 − 0.335i)11-s + (−0.571 + 0.0845i)12-s + (−0.0752 − 0.0752i)13-s + (−0.477 + 0.240i)14-s − 0.258·15-s + (−0.289 − 0.957i)16-s + 0.180·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.633 - 0.773i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.633 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.654445 + 1.38268i\)
\(L(\frac12)\) \(\approx\) \(0.654445 + 1.38268i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.635 - 1.26i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 + (-1.11 + 1.11i)T - 11iT^{2} \)
13 \( 1 + (0.271 + 0.271i)T + 13iT^{2} \)
17 \( 1 - 0.744T + 17T^{2} \)
19 \( 1 + (-5.21 - 5.21i)T + 19iT^{2} \)
23 \( 1 + 4.76iT - 23T^{2} \)
29 \( 1 + (1.21 + 1.21i)T + 29iT^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 + (5.32 - 5.32i)T - 37iT^{2} \)
41 \( 1 + 7.33iT - 41T^{2} \)
43 \( 1 + (-6.78 + 6.78i)T - 43iT^{2} \)
47 \( 1 + 0.735T + 47T^{2} \)
53 \( 1 + (9.55 - 9.55i)T - 53iT^{2} \)
59 \( 1 + (-1.62 + 1.62i)T - 59iT^{2} \)
61 \( 1 + (5.70 + 5.70i)T + 61iT^{2} \)
67 \( 1 + (-5.59 - 5.59i)T + 67iT^{2} \)
71 \( 1 + 8.60iT - 71T^{2} \)
73 \( 1 + 4.28iT - 73T^{2} \)
79 \( 1 + 1.01T + 79T^{2} \)
83 \( 1 + (-1.68 - 1.68i)T + 83iT^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 + 16.9T + 97T^{2} \)
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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

−12.42010276128107023544769963060, −11.86238492630281475036735114931, −10.42931205886520501620000891420, −9.321039066774243775236008581528, −8.402839685181807939687812975720, −7.58000020422975030036297263322, −6.35006912238156785006350768444, −5.31283829595860746770370644470, −4.03476116206543014098681770947, −2.95538881032239255815899046524, 1.20257994433295965988366961679, 2.92005587483305609350615346357, 4.11966878623242114374011044104, 5.25740023395859639896031838072, 6.76133441868883457879465868738, 7.87751520378398968998170624036, 9.157036239478735546931611750122, 9.811781125732730687015677134133, 11.11292034295606834696471631938, 11.82111427378720817371869976524

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