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Properties

Label 2-3920-5.2-c0-0-1
Degree $2$
Conductor $3920$
Sign $0.973 + 0.229i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  + (0.707 − 0.707i)3-s + (−0.707 + 0.707i)5-s − 11-s + (0.707 − 0.707i)13-s + 1.00i·15-s + (0.707 + 0.707i)17-s − 1.41i·19-s − 1.00i·25-s + (0.707 + 0.707i)27-s + i·29-s + 1.41·31-s + (−0.707 + 0.707i)33-s − 1.00i·39-s + 1.41·41-s + (1 − i)43-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.707 + 0.707i)5-s − 11-s + (0.707 − 0.707i)13-s + 1.00i·15-s + (0.707 + 0.707i)17-s − 1.41i·19-s − 1.00i·25-s + (0.707 + 0.707i)27-s + i·29-s + 1.41·31-s + (−0.707 + 0.707i)33-s − 1.00i·39-s + 1.41·41-s + (1 − i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (3137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.468762194\)
\(L(\frac12)\) \(\approx\) \(1.468762194\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 \)
good3 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-1 - i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
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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

−8.319771090091754007081840682722, −7.901550860273481960176424779520, −7.35132355595854840613288116389, −6.63168260153703269630829601685, −5.74004286855626282028983112837, −4.83386695180336351868824776755, −3.81428946703503250547807303639, −2.86033242427510039714820921232, −2.53954315053312156510090749268, −1.04361465500690467668544771598, 1.02196180870556659317111013430, 2.47316934907743895496110657007, 3.37926226028134248985625136621, 4.08600791812452125642693982464, 4.64943131217025843593529847374, 5.61527489781115794715216122694, 6.37667653346239747941976201394, 7.63178226402554640364779867257, 7.981918970455919736687726187598, 8.638433168516699370497485985722

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