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Properties

Label 2-240-80.43-c1-0-1
Degree $2$
Conductor $240$
Sign $-0.909 - 0.416i$
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  + (−1.41 + 0.0770i)2-s + i·3-s + (1.98 − 0.217i)4-s + (−0.658 − 2.13i)5-s + (−0.0770 − 1.41i)6-s + (−3.54 + 3.54i)7-s + (−2.79 + 0.460i)8-s − 9-s + (1.09 + 2.96i)10-s + (−0.707 + 0.707i)11-s + (0.217 + 1.98i)12-s + 1.18·13-s + (4.73 − 5.28i)14-s + (2.13 − 0.658i)15-s + (3.90 − 0.865i)16-s + (−2.63 + 2.63i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0544i)2-s + 0.577i·3-s + (0.994 − 0.108i)4-s + (−0.294 − 0.955i)5-s + (−0.0314 − 0.576i)6-s + (−1.34 + 1.34i)7-s + (−0.986 + 0.162i)8-s − 0.333·9-s + (0.346 + 0.938i)10-s + (−0.213 + 0.213i)11-s + (0.0628 + 0.573i)12-s + 0.329·13-s + (1.26 − 1.41i)14-s + (0.551 − 0.170i)15-s + (0.976 − 0.216i)16-s + (−0.639 + 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.909 - 0.416i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.909 - 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0628146 + 0.288126i\)
\(L(\frac12)\) \(\approx\) \(0.0628146 + 0.288126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0770i)T \)
3 \( 1 - iT \)
5 \( 1 + (0.658 + 2.13i)T \)
good7 \( 1 + (3.54 - 3.54i)T - 7iT^{2} \)
11 \( 1 + (0.707 - 0.707i)T - 11iT^{2} \)
13 \( 1 - 1.18T + 13T^{2} \)
17 \( 1 + (2.63 - 2.63i)T - 17iT^{2} \)
19 \( 1 + (5.21 - 5.21i)T - 19iT^{2} \)
23 \( 1 + (1.86 + 1.86i)T + 23iT^{2} \)
29 \( 1 + (2.17 + 2.17i)T + 29iT^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 - 0.910T + 37T^{2} \)
41 \( 1 - 8.26iT - 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + (5.06 + 5.06i)T + 47iT^{2} \)
53 \( 1 - 3.52iT - 53T^{2} \)
59 \( 1 + (-10.2 - 10.2i)T + 59iT^{2} \)
61 \( 1 + (-4.49 + 4.49i)T - 61iT^{2} \)
67 \( 1 - 1.27T + 67T^{2} \)
71 \( 1 + 3.56T + 71T^{2} \)
73 \( 1 + (-2.47 + 2.47i)T - 73iT^{2} \)
79 \( 1 + 3.89T + 79T^{2} \)
83 \( 1 - 9.99iT - 83T^{2} \)
89 \( 1 - 5.16T + 89T^{2} \)
97 \( 1 + (6.87 - 6.87i)T - 97iT^{2} \)
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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

−12.44465087107071524052431364047, −11.53252714787979355364816692810, −10.30947698656499477888842375688, −9.514509454871814183901482203722, −8.763487223303428230757595923132, −8.108863249121362818883263967560, −6.36359435559121511227151155087, −5.69170643536542582561200743489, −3.91509074492173030918007059729, −2.31432932601574191033226259763, 0.29987443936630706270650607078, 2.58830316638720076462121887582, 3.74360535455323142687574581926, 6.25310939225400003826551712517, 6.92191591689103371927788355905, 7.48764438548892128279475996541, 8.814055345698279404705287511727, 9.885395405015957101861210544454, 10.78061787068271084940615861557, 11.27975195471458061011553849618

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