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Properties

Label 2-240-16.13-c1-0-6
Degree $2$
Conductor $240$
Sign $0.272 - 0.962i$
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  + (1.19 + 0.750i)2-s + (−0.707 − 0.707i)3-s + (0.872 + 1.79i)4-s + (−0.707 + 0.707i)5-s + (−0.316 − 1.37i)6-s + 3.79i·7-s + (−0.306 + 2.81i)8-s + 1.00i·9-s + (−1.37 + 0.316i)10-s + (3.08 − 3.08i)11-s + (0.656 − 1.88i)12-s + (1.54 + 1.54i)13-s + (−2.85 + 4.55i)14-s + 1.00·15-s + (−2.47 + 3.13i)16-s + 4.32·17-s + ⋯
L(s)  = 1  + (0.847 + 0.531i)2-s + (−0.408 − 0.408i)3-s + (0.436 + 0.899i)4-s + (−0.316 + 0.316i)5-s + (−0.129 − 0.562i)6-s + 1.43i·7-s + (−0.108 + 0.994i)8-s + 0.333i·9-s + (−0.435 + 0.100i)10-s + (0.930 − 0.930i)11-s + (0.189 − 0.545i)12-s + (0.428 + 0.428i)13-s + (−0.762 + 1.21i)14-s + 0.258·15-s + (−0.619 + 0.784i)16-s + 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.272 - 0.962i$
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: \((2,\ 240,\ (\ :1/2),\ 0.272 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34479 + 1.01705i\)
\(L(\frac12)\) \(\approx\) \(1.34479 + 1.01705i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 - 0.750i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - 3.79iT - 7T^{2} \)
11 \( 1 + (-3.08 + 3.08i)T - 11iT^{2} \)
13 \( 1 + (-1.54 - 1.54i)T + 13iT^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 + (5.37 + 5.37i)T + 19iT^{2} \)
23 \( 1 + 3.91iT - 23T^{2} \)
29 \( 1 + (-1.84 - 1.84i)T + 29iT^{2} \)
31 \( 1 + 9.52T + 31T^{2} \)
37 \( 1 + (-4.55 + 4.55i)T - 37iT^{2} \)
41 \( 1 - 0.580iT - 41T^{2} \)
43 \( 1 + (-0.994 + 0.994i)T - 43iT^{2} \)
47 \( 1 - 2.22T + 47T^{2} \)
53 \( 1 + (-4.80 + 4.80i)T - 53iT^{2} \)
59 \( 1 + (-7.26 + 7.26i)T - 59iT^{2} \)
61 \( 1 + (-0.301 - 0.301i)T + 61iT^{2} \)
67 \( 1 + (-6.97 - 6.97i)T + 67iT^{2} \)
71 \( 1 - 0.585iT - 71T^{2} \)
73 \( 1 - 11.9iT - 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + (11.1 + 11.1i)T + 83iT^{2} \)
89 \( 1 + 12.9iT - 89T^{2} \)
97 \( 1 + 6.78T + 97T^{2} \)
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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

−12.39933066111044741374145306083, −11.55812727168290657140179534657, −10.99498759882929208985160876944, −8.994351372507227690772366807273, −8.366833581591688426717565859314, −6.96420341790981559548720764318, −6.18620753578543423298867729250, −5.34809552525823730825027354006, −3.86163159311429529408915137973, −2.45662202213569989670902939082, 1.32001251793413450927641591194, 3.75717148332990650749088661838, 4.17216167361828779940525313422, 5.51588590384656732308961896763, 6.68288399943641051699576412733, 7.75978686251092392365229515220, 9.491609322964948714315873206380, 10.28617814627696231090570083939, 10.98834245270875820448674546864, 12.04438054444561944502762790960

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