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Properties

Label 2-3520-440.349-c0-0-2
Degree $2$
Conductor $3520$
Sign $0.920 + 0.390i$
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  + (0.809 + 0.587i)5-s + (0.587 − 1.80i)7-s + (0.809 − 0.587i)9-s + (−0.951 − 0.309i)11-s + (0.690 + 0.951i)13-s + (0.363 + 1.11i)19-s + 0.618i·23-s + (0.309 + 0.951i)25-s + (1.53 − 1.11i)35-s + (0.5 − 1.53i)37-s + (−1.80 + 0.587i)41-s + 45-s + (1.53 − 0.5i)47-s + (−2.11 − 1.53i)49-s + (0.5 − 0.363i)53-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)5-s + (0.587 − 1.80i)7-s + (0.809 − 0.587i)9-s + (−0.951 − 0.309i)11-s + (0.690 + 0.951i)13-s + (0.363 + 1.11i)19-s + 0.618i·23-s + (0.309 + 0.951i)25-s + (1.53 − 1.11i)35-s + (0.5 − 1.53i)37-s + (−1.80 + 0.587i)41-s + 45-s + (1.53 − 0.5i)47-s + (−2.11 − 1.53i)49-s + (0.5 − 0.363i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.920 + 0.390i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ 0.920 + 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.673553410\)
\(L(\frac12)\) \(\approx\) \(1.673553410\)
\(L(1)\) 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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.951 + 0.309i)T \)
good3 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 - 0.618iT - T^{2} \)
29 \( 1 + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + 0.618T + T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−8.708991565325419957629718537305, −7.68242847231942615138292443465, −7.29405098543058058467303623190, −6.57379236763996831988261824026, −5.80718836649660275768000981074, −4.84311045653000274219134149131, −3.90656357107299453710256194397, −3.44500958553286535428216533616, −1.94410431995240784615811881094, −1.18370455312324631337297369549, 1.39295930296836959393308209396, 2.34473772972143154755191844756, 2.90553225715869771878895300054, 4.60051241811588135340009764360, 5.07251534307986526443706616815, 5.62391385764842093682089964530, 6.36706251859616300651514546320, 7.47342617393161759536669347629, 8.273326240036432605885408296606, 8.699427055656215141127633980189

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