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Properties

Label 2-240-80.29-c1-0-0
Degree $2$
Conductor $240$
Sign $-0.661 + 0.749i$
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  + (0.345 + 1.37i)2-s + (−0.707 + 0.707i)3-s + (−1.76 + 0.946i)4-s + (−0.561 − 2.16i)5-s + (−1.21 − 0.725i)6-s − 4.51·7-s + (−1.90 − 2.08i)8-s − 1.00i·9-s + (2.77 − 1.51i)10-s + (−3.44 + 3.44i)11-s + (0.576 − 1.91i)12-s + (0.113 − 0.113i)13-s + (−1.55 − 6.19i)14-s + (1.92 + 1.13i)15-s + (2.20 − 3.33i)16-s + 5.03i·17-s + ⋯
L(s)  = 1  + (0.244 + 0.969i)2-s + (−0.408 + 0.408i)3-s + (−0.880 + 0.473i)4-s + (−0.251 − 0.967i)5-s + (−0.495 − 0.296i)6-s − 1.70·7-s + (−0.673 − 0.738i)8-s − 0.333i·9-s + (0.877 − 0.479i)10-s + (−1.03 + 1.03i)11-s + (0.166 − 0.552i)12-s + (0.0315 − 0.0315i)13-s + (−0.416 − 1.65i)14-s + (0.497 + 0.292i)15-s + (0.551 − 0.833i)16-s + 1.22i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.661 + 0.749i$
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: \((2,\ 240,\ (\ :1/2),\ -0.661 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0820432 - 0.181926i\)
\(L(\frac12)\) \(\approx\) \(0.0820432 - 0.181926i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.345 - 1.37i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.561 + 2.16i)T \)
good7 \( 1 + 4.51T + 7T^{2} \)
11 \( 1 + (3.44 - 3.44i)T - 11iT^{2} \)
13 \( 1 + (-0.113 + 0.113i)T - 13iT^{2} \)
17 \( 1 - 5.03iT - 17T^{2} \)
19 \( 1 + (0.992 + 0.992i)T + 19iT^{2} \)
23 \( 1 - 8.00T + 23T^{2} \)
29 \( 1 + (1.01 + 1.01i)T + 29iT^{2} \)
31 \( 1 + 6.42T + 31T^{2} \)
37 \( 1 + (1.63 + 1.63i)T + 37iT^{2} \)
41 \( 1 - 3.35iT - 41T^{2} \)
43 \( 1 + (5.68 + 5.68i)T + 43iT^{2} \)
47 \( 1 - 9.10iT - 47T^{2} \)
53 \( 1 + (3.27 + 3.27i)T + 53iT^{2} \)
59 \( 1 + (5.30 - 5.30i)T - 59iT^{2} \)
61 \( 1 + (5.87 + 5.87i)T + 61iT^{2} \)
67 \( 1 + (-1.87 + 1.87i)T - 67iT^{2} \)
71 \( 1 + 0.635iT - 71T^{2} \)
73 \( 1 + 6.14T + 73T^{2} \)
79 \( 1 - 1.76T + 79T^{2} \)
83 \( 1 + (-6.39 + 6.39i)T - 83iT^{2} \)
89 \( 1 - 0.579iT - 89T^{2} \)
97 \( 1 - 15.2iT - 97T^{2} \)
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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

−12.88720528675821927403852828084, −12.31623545780633777719142578598, −10.59701006347567638676553004509, −9.577240728731433012589279834605, −8.944082400939262940858354431421, −7.65742709554653206532235553951, −6.61710508524053017631133069344, −5.56757236303365889143841303206, −4.61326439123376189619133773994, −3.42569437858897153475121106809, 0.14880834824074971559494195170, 2.78644008064039313696558650836, 3.39929404925038268261512875001, 5.28366148909343370139895619221, 6.32706277985959528426329045372, 7.33830441891278496287731582230, 8.875276757143042032791261528970, 9.924187868018854336353128346355, 10.75379261078266460272842637501, 11.40484772511433188417094098269

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