Properties

Label 2-240-12.11-c1-0-5
Degree $2$
Conductor $240$
Sign $0.707 + 0.707i$
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  + (1.22 − 1.22i)3-s + i·5-s − 2.44i·7-s − 2.99i·9-s + 4.89·11-s − 2·13-s + (1.22 + 1.22i)15-s + 6i·17-s − 4.89i·19-s + (−2.99 − 2.99i)21-s − 2.44·23-s − 25-s + (−3.67 − 3.67i)27-s + 9.79i·31-s + (5.99 − 5.99i)33-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + 0.447i·5-s − 0.925i·7-s − 0.999i·9-s + 1.47·11-s − 0.554·13-s + (0.316 + 0.316i)15-s + 1.45i·17-s − 1.12i·19-s + (−0.654 − 0.654i)21-s − 0.510·23-s − 0.200·25-s + (−0.707 − 0.707i)27-s + 1.75i·31-s + (1.04 − 1.04i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43925 - 0.596157i\)
\(L(\frac12)\) \(\approx\) \(1.43925 - 0.596157i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 - iT \)
good7 \( 1 + 2.44iT - 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 4.89iT - 19T^{2} \)
23 \( 1 + 2.44T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 9.79iT - 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 - 2.44iT - 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 9.79T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 7.34iT - 67T^{2} \)
71 \( 1 - 4.89T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 - 4.89iT - 79T^{2} \)
83 \( 1 - 7.34T + 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−12.12923503844817537246235427186, −11.10912735253792462343126601525, −10.00821849595029717031468043469, −9.023836039567602619593387196662, −7.993728322752909466097490185745, −6.93695740277744433637739623266, −6.37378796084654952220164609089, −4.31662415329425293006140228648, −3.22926287592756244948709443959, −1.51124405817603374519978299934, 2.17575853976528079729760166823, 3.67521549888056452578309702761, 4.81455572398157720119299431592, 5.97291331918668535864685692226, 7.50229089351175350291436840500, 8.588896188842406844999798465121, 9.375123936841643347833992673525, 9.926897748076442001144584483734, 11.52199956625468574143960049595, 12.04221196389464327936664572000

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