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Properties

Label 2-240-80.69-c1-0-6
Degree $2$
Conductor $240$
Sign $0.437 - 0.899i$
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  + (−1.06 + 0.933i)2-s + (0.707 + 0.707i)3-s + (0.256 − 1.98i)4-s + (−1.86 − 1.24i)5-s + (−1.41 − 0.0909i)6-s + 1.58·7-s + (1.57 + 2.34i)8-s + 1.00i·9-s + (3.13 − 0.418i)10-s + (3.92 + 3.92i)11-s + (1.58 − 1.22i)12-s + (3.10 + 3.10i)13-s + (−1.68 + 1.48i)14-s + (−0.438 − 2.19i)15-s + (−3.86 − 1.01i)16-s + 1.48i·17-s + ⋯
L(s)  = 1  + (−0.751 + 0.660i)2-s + (0.408 + 0.408i)3-s + (0.128 − 0.991i)4-s + (−0.831 − 0.554i)5-s + (−0.576 − 0.0371i)6-s + 0.600·7-s + (0.558 + 0.829i)8-s + 0.333i·9-s + (0.991 − 0.132i)10-s + (1.18 + 1.18i)11-s + (0.457 − 0.352i)12-s + (0.861 + 0.861i)13-s + (−0.451 + 0.396i)14-s + (−0.113 − 0.566i)15-s + (−0.967 − 0.254i)16-s + 0.359i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.830348 + 0.519338i\)
\(L(\frac12)\) \(\approx\) \(0.830348 + 0.519338i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.06 - 0.933i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (1.86 + 1.24i)T \)
good7 \( 1 - 1.58T + 7T^{2} \)
11 \( 1 + (-3.92 - 3.92i)T + 11iT^{2} \)
13 \( 1 + (-3.10 - 3.10i)T + 13iT^{2} \)
17 \( 1 - 1.48iT - 17T^{2} \)
19 \( 1 + (-4.94 + 4.94i)T - 19iT^{2} \)
23 \( 1 + 6.61T + 23T^{2} \)
29 \( 1 + (-4.42 + 4.42i)T - 29iT^{2} \)
31 \( 1 + 1.50T + 31T^{2} \)
37 \( 1 + (2.14 - 2.14i)T - 37iT^{2} \)
41 \( 1 - 6.84iT - 41T^{2} \)
43 \( 1 + (-0.322 + 0.322i)T - 43iT^{2} \)
47 \( 1 + 13.3iT - 47T^{2} \)
53 \( 1 + (0.931 - 0.931i)T - 53iT^{2} \)
59 \( 1 + (-1.14 - 1.14i)T + 59iT^{2} \)
61 \( 1 + (-2.67 + 2.67i)T - 61iT^{2} \)
67 \( 1 + (5.43 + 5.43i)T + 67iT^{2} \)
71 \( 1 + 2.26iT - 71T^{2} \)
73 \( 1 + 5.27T + 73T^{2} \)
79 \( 1 + 6.52T + 79T^{2} \)
83 \( 1 + (0.973 + 0.973i)T + 83iT^{2} \)
89 \( 1 + 6.83iT - 89T^{2} \)
97 \( 1 - 15.8iT - 97T^{2} \)
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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

−11.86172112609428183095584877381, −11.43459400477162203203841763931, −10.04108444860454361532128787032, −9.186256721562367966803565816527, −8.462429145248475244537361972725, −7.54465904541142528077165111970, −6.52032246913440495175238927144, −4.89916651815893929093710431186, −4.06844400826628862746234557193, −1.60382806776286510684919672608, 1.21790519721917841456281585919, 3.14375085734061564941634027913, 3.88619380214943419367444177063, 6.09948644482808055416672831511, 7.40150619402082217797075424324, 8.164482273876539887343471668508, 8.820606347409107780858961735701, 10.15400818541482188895343578849, 11.13787176031352738643741062485, 11.74849129147695585409348106004

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