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Properties

Label 2-240-20.7-c1-0-0
Degree $2$
Conductor $240$
Sign $-0.189 - 0.981i$
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  + (−0.707 − 0.707i)3-s + (−0.224 + 2.22i)5-s + (−3.14 + 3.14i)7-s + 1.00i·9-s − 1.41i·11-s + (−2.44 + 2.44i)13-s + (1.73 − 1.41i)15-s + (4.44 + 4.44i)17-s − 0.635·19-s + 4.44·21-s + (2.04 + 2.04i)23-s + (−4.89 − i)25-s + (0.707 − 0.707i)27-s − 9.34i·29-s + 8.48i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.100 + 0.994i)5-s + (−1.18 + 1.18i)7-s + 0.333i·9-s − 0.426i·11-s + (−0.679 + 0.679i)13-s + (0.447 − 0.365i)15-s + (1.07 + 1.07i)17-s − 0.145·19-s + 0.970·21-s + (0.427 + 0.427i)23-s + (−0.979 − 0.200i)25-s + (0.136 − 0.136i)27-s − 1.73i·29-s + 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.189 - 0.981i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.474149 + 0.574667i\)
\(L(\frac12)\) \(\approx\) \(0.474149 + 0.574667i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.224 - 2.22i)T \)
good7 \( 1 + (3.14 - 3.14i)T - 7iT^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + (2.44 - 2.44i)T - 13iT^{2} \)
17 \( 1 + (-4.44 - 4.44i)T + 17iT^{2} \)
19 \( 1 + 0.635T + 19T^{2} \)
23 \( 1 + (-2.04 - 2.04i)T + 23iT^{2} \)
29 \( 1 + 9.34iT - 29T^{2} \)
31 \( 1 - 8.48iT - 31T^{2} \)
37 \( 1 + (1.55 + 1.55i)T + 37iT^{2} \)
41 \( 1 + 1.10T + 41T^{2} \)
43 \( 1 + (0.635 + 0.635i)T + 43iT^{2} \)
47 \( 1 + (-5.51 + 5.51i)T - 47iT^{2} \)
53 \( 1 + (2 - 2i)T - 53iT^{2} \)
59 \( 1 - 9.89T + 59T^{2} \)
61 \( 1 - 2.89T + 61T^{2} \)
67 \( 1 + (6.29 - 6.29i)T - 67iT^{2} \)
71 \( 1 - 12.5iT - 71T^{2} \)
73 \( 1 + (-3 + 3i)T - 73iT^{2} \)
79 \( 1 - 9.75T + 79T^{2} \)
83 \( 1 + (1.41 + 1.41i)T + 83iT^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
97 \( 1 + (-3 - 3i)T + 97iT^{2} \)
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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

−12.21643520587958325911108266444, −11.68689062009718927080616214482, −10.44747086845555357228229509651, −9.663769078700010749322949143404, −8.472581201823473482447133041698, −7.19324425065220178784821244601, −6.31461099282144113542424009213, −5.56344323208726047995933905993, −3.57785749786600299151655869036, −2.38763336143636504048320041018, 0.62422920614802035164289667134, 3.28610305669540034216297133961, 4.51554978778315875007965161593, 5.48820259733349925057346778014, 6.88684801507489430442515566875, 7.77790381119320645557536700955, 9.298765123466925082262259135829, 9.866292789318367400419197281195, 10.72943807823724504351698745555, 12.06744693909148806822189239792

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