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Properties

Label 2-3520-55.54-c0-0-7
Degree $2$
Conductor $3520$
Sign $-0.5 + 0.866i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s + (−0.5 + 0.866i)5-s − 1.99·9-s + 11-s + (1.49 + 0.866i)15-s − 1.73i·23-s + (−0.499 − 0.866i)25-s + 1.73i·27-s + 31-s − 1.73i·33-s − 1.73i·37-s + (0.999 − 1.73i)45-s − 49-s + (−0.5 + 0.866i)55-s + 59-s + ⋯
L(s)  = 1  − 1.73i·3-s + (−0.5 + 0.866i)5-s − 1.99·9-s + 11-s + (1.49 + 0.866i)15-s − 1.73i·23-s + (−0.499 − 0.866i)25-s + 1.73i·27-s + 31-s − 1.73i·33-s − 1.73i·37-s + (0.999 − 1.73i)45-s − 49-s + (−0.5 + 0.866i)55-s + 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.5 + 0.866i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.058630050\)
\(L(\frac12)\) \(\approx\) \(1.058630050\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 - T \)
good3 \( 1 + 1.73iT - T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.73iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−8.331043546101537887068527858373, −7.59749879773738675976605309839, −7.02429305130262922309361396371, −6.40027671107294541287041652865, −6.02680022567003841989920176188, −4.64775630912575104777832088094, −3.62597990196896818456932551952, −2.67715785801943253071926931538, −1.94097323561496229679135522449, −0.67267543587159371333508382746, 1.35681655633745390270161755919, 3.03809788939863623949091890353, 3.76950079489309455784339537086, 4.35135847694213903017854313937, 5.03543511034848864512015136990, 5.68429494620148868538969595951, 6.66598325762338124525744882663, 7.79551053844817427088859375420, 8.518319594994623319076389768314, 9.075751851269258745975030761993

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