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Properties

Label 2-240-16.5-c1-0-10
Degree $2$
Conductor $240$
Sign $-0.516 + 0.856i$
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  + (−0.491 − 1.32i)2-s + (0.707 − 0.707i)3-s + (−1.51 + 1.30i)4-s + (0.707 + 0.707i)5-s + (−1.28 − 0.589i)6-s − 3.46i·7-s + (2.47 + 1.36i)8-s − 1.00i·9-s + (0.589 − 1.28i)10-s + (−2.79 − 2.79i)11-s + (−0.149 + 1.99i)12-s + (2.41 − 2.41i)13-s + (−4.59 + 1.70i)14-s + 1.00·15-s + (0.596 − 3.95i)16-s + 0.598·17-s + ⋯
L(s)  = 1  + (−0.347 − 0.937i)2-s + (0.408 − 0.408i)3-s + (−0.757 + 0.652i)4-s + (0.316 + 0.316i)5-s + (−0.524 − 0.240i)6-s − 1.31i·7-s + (0.875 + 0.483i)8-s − 0.333i·9-s + (0.186 − 0.406i)10-s + (−0.843 − 0.843i)11-s + (−0.0431 + 0.575i)12-s + (0.669 − 0.669i)13-s + (−1.22 + 0.455i)14-s + 0.258·15-s + (0.149 − 0.988i)16-s + 0.145·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.516 + 0.856i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.516 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.546057 - 0.966585i\)
\(L(\frac12)\) \(\approx\) \(0.546057 - 0.966585i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.491 + 1.32i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 + (2.79 + 2.79i)T + 11iT^{2} \)
13 \( 1 + (-2.41 + 2.41i)T - 13iT^{2} \)
17 \( 1 - 0.598T + 17T^{2} \)
19 \( 1 + (-1.22 + 1.22i)T - 19iT^{2} \)
23 \( 1 - 3.77iT - 23T^{2} \)
29 \( 1 + (7.33 - 7.33i)T - 29iT^{2} \)
31 \( 1 - 6.08T + 31T^{2} \)
37 \( 1 + (-7.71 - 7.71i)T + 37iT^{2} \)
41 \( 1 + 6.67iT - 41T^{2} \)
43 \( 1 + (5.43 + 5.43i)T + 43iT^{2} \)
47 \( 1 - 6.63T + 47T^{2} \)
53 \( 1 + (-6.76 - 6.76i)T + 53iT^{2} \)
59 \( 1 + (-6.80 - 6.80i)T + 59iT^{2} \)
61 \( 1 + (3.65 - 3.65i)T - 61iT^{2} \)
67 \( 1 + (-1.30 + 1.30i)T - 67iT^{2} \)
71 \( 1 + 4.48iT - 71T^{2} \)
73 \( 1 - 4.50iT - 73T^{2} \)
79 \( 1 - 0.465T + 79T^{2} \)
83 \( 1 + (9.66 - 9.66i)T - 83iT^{2} \)
89 \( 1 + 4.19iT - 89T^{2} \)
97 \( 1 - 5.24T + 97T^{2} \)
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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

−11.61352201676793914718851072406, −10.69947169674100268770075450040, −10.18475728432109442975733244269, −8.946217244114387277875008200730, −7.949350258526496527082903649658, −7.16400672766210997065003172816, −5.50124562771955711482060494403, −3.82766530954911910425879466623, −2.88182294778484834807918570630, −1.07012744832584877207540409574, 2.23344012988583450877679390084, 4.31473931541000755526209001626, 5.39466849962271528601381290275, 6.27882765029352654982955283698, 7.74328393982440502131641396477, 8.545539914484379174899495637876, 9.424131529130278502567615345266, 10.02995714846100953778063045896, 11.41750093896585151647856454170, 12.72398696989681404774657554520

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