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Properties

Label 2-4020-5.4-c1-0-37
Degree $2$
Conductor $4020$
Sign $0.990 - 0.141i$
Analytic cond. $32.0998$
Root an. cond. $5.66567$
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  + i·3-s + (2.21 − 0.315i)5-s + 0.0918i·7-s − 9-s − 3.06·11-s − 5.35i·13-s + (0.315 + 2.21i)15-s + 2.02i·17-s + 7.88·19-s − 0.0918·21-s + 5.50i·23-s + (4.80 − 1.39i)25-s − i·27-s − 2.34·29-s − 1.86·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.990 − 0.141i)5-s + 0.0347i·7-s − 0.333·9-s − 0.923·11-s − 1.48i·13-s + (0.0814 + 0.571i)15-s + 0.492i·17-s + 1.80·19-s − 0.0200·21-s + 1.14i·23-s + (0.960 − 0.279i)25-s − 0.192i·27-s − 0.435·29-s − 0.335·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.990 - 0.141i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (1609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.990 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.245970564\)
\(L(\frac12)\) \(\approx\) \(2.245970564\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (-2.21 + 0.315i)T \)
67 \( 1 - iT \)
good7 \( 1 - 0.0918iT - 7T^{2} \)
11 \( 1 + 3.06T + 11T^{2} \)
13 \( 1 + 5.35iT - 13T^{2} \)
17 \( 1 - 2.02iT - 17T^{2} \)
19 \( 1 - 7.88T + 19T^{2} \)
23 \( 1 - 5.50iT - 23T^{2} \)
29 \( 1 + 2.34T + 29T^{2} \)
31 \( 1 + 1.86T + 31T^{2} \)
37 \( 1 + 6.88iT - 37T^{2} \)
41 \( 1 + 5.27T + 41T^{2} \)
43 \( 1 - 6.19iT - 43T^{2} \)
47 \( 1 + 3.51iT - 47T^{2} \)
53 \( 1 - 1.50iT - 53T^{2} \)
59 \( 1 - 7.85T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + 6.48iT - 73T^{2} \)
79 \( 1 - 8.72T + 79T^{2} \)
83 \( 1 + 3.43iT - 83T^{2} \)
89 \( 1 - 4.92T + 89T^{2} \)
97 \( 1 - 3.65iT - 97T^{2} \)
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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

−8.467201346295137872455597357499, −7.79537312366796050103623129187, −7.09954530035153848008023098724, −5.86588320455952925695819276656, −5.42326641902881677285774995897, −5.08226337726605185505685732733, −3.68378498917271482507744842828, −3.05054759036481549937507292321, −2.09670001557287879230279538323, −0.812959965645074319168758609190, 0.930621101673477384771255784771, 2.04770441183105429666464355002, 2.64433019345373162613132955043, 3.70685672979882029831063929323, 5.01676815549585183268366425966, 5.33846105950175498337387092027, 6.38569337090952876398296511585, 6.91063765796460228030642386795, 7.53912559928598752586365227139, 8.482642456440773516194242393028

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