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Properties

Label 2-240-80.29-c1-0-7
Degree $2$
Conductor $240$
Sign $0.988 + 0.148i$
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  + (−1.20 + 0.742i)2-s + (−0.707 + 0.707i)3-s + (0.898 − 1.78i)4-s + (−2.06 − 0.860i)5-s + (0.326 − 1.37i)6-s + 0.707·7-s + (0.244 + 2.81i)8-s − 1.00i·9-s + (3.12 − 0.495i)10-s + (1.79 − 1.79i)11-s + (0.628 + 1.89i)12-s + (3.86 − 3.86i)13-s + (−0.851 + 0.524i)14-s + (2.06 − 0.850i)15-s + (−2.38 − 3.21i)16-s − 0.244i·17-s + ⋯
L(s)  = 1  + (−0.851 + 0.524i)2-s + (−0.408 + 0.408i)3-s + (0.449 − 0.893i)4-s + (−0.922 − 0.384i)5-s + (0.133 − 0.561i)6-s + 0.267·7-s + (0.0863 + 0.996i)8-s − 0.333i·9-s + (0.987 − 0.156i)10-s + (0.542 − 0.542i)11-s + (0.181 + 0.548i)12-s + (1.07 − 1.07i)13-s + (−0.227 + 0.140i)14-s + (0.533 − 0.219i)15-s + (−0.596 − 0.802i)16-s − 0.0593i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.988 + 0.148i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.988 + 0.148i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.660409 - 0.0492920i\)
\(L(\frac12)\) \(\approx\) \(0.660409 - 0.0492920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.742i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (2.06 + 0.860i)T \)
good7 \( 1 - 0.707T + 7T^{2} \)
11 \( 1 + (-1.79 + 1.79i)T - 11iT^{2} \)
13 \( 1 + (-3.86 + 3.86i)T - 13iT^{2} \)
17 \( 1 + 0.244iT - 17T^{2} \)
19 \( 1 + (-1.53 - 1.53i)T + 19iT^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + (4.89 + 4.89i)T + 29iT^{2} \)
31 \( 1 - 7.60T + 31T^{2} \)
37 \( 1 + (8.47 + 8.47i)T + 37iT^{2} \)
41 \( 1 - 2.12iT - 41T^{2} \)
43 \( 1 + (0.684 + 0.684i)T + 43iT^{2} \)
47 \( 1 + 4.47iT - 47T^{2} \)
53 \( 1 + (-1.47 - 1.47i)T + 53iT^{2} \)
59 \( 1 + (-5.86 + 5.86i)T - 59iT^{2} \)
61 \( 1 + (-0.0537 - 0.0537i)T + 61iT^{2} \)
67 \( 1 + (7.85 - 7.85i)T - 67iT^{2} \)
71 \( 1 + 2.08iT - 71T^{2} \)
73 \( 1 - 9.69T + 73T^{2} \)
79 \( 1 + 7.34T + 79T^{2} \)
83 \( 1 + (6.80 - 6.80i)T - 83iT^{2} \)
89 \( 1 + 3.07iT - 89T^{2} \)
97 \( 1 + 1.39iT - 97T^{2} \)
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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

−11.62366999291647155783141343373, −11.15163122979972349089000587513, −10.19597815651876057565212401366, −8.950559582849331348957200332307, −8.323122789195239361730858682367, −7.29025896509972765583669066602, −6.03415392374771693883846725245, −5.04425499950965648330743211493, −3.53690639685039804350203296328, −0.880913168175172710937651558449, 1.42289267192764187594513476245, 3.26408742373867991986140773146, 4.55473460642466664166867915291, 6.62660717050320494007325853197, 7.15223383202169394450726861653, 8.336179446136585275679656523623, 9.143082186176398772756017266982, 10.47116103304358765361233937875, 11.36776459707274273109767195165, 11.72724895325712875770700513256

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