Properties

Label 2-4020-5.4-c1-0-18
Degree $2$
Conductor $4020$
Sign $0.928 + 0.372i$
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 + (−2.07 − 0.832i)5-s − 0.944i·7-s − 9-s − 5.62·11-s + 4.06i·13-s + (−0.832 + 2.07i)15-s + 2.26i·17-s − 2.66·19-s − 0.944·21-s − 4.72i·23-s + (3.61 + 3.45i)25-s + i·27-s − 3.13·29-s + 3.52·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.928 − 0.372i)5-s − 0.357i·7-s − 0.333·9-s − 1.69·11-s + 1.12i·13-s + (−0.214 + 0.535i)15-s + 0.548i·17-s − 0.612·19-s − 0.206·21-s − 0.984i·23-s + (0.723 + 0.690i)25-s + 0.192i·27-s − 0.582·29-s + 0.633·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.928 + 0.372i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.928 + 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9117489764\)
\(L(\frac12)\) \(\approx\) \(0.9117489764\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 + 0.832i)T \)
67 \( 1 - iT \)
good7 \( 1 + 0.944iT - 7T^{2} \)
11 \( 1 + 5.62T + 11T^{2} \)
13 \( 1 - 4.06iT - 13T^{2} \)
17 \( 1 - 2.26iT - 17T^{2} \)
19 \( 1 + 2.66T + 19T^{2} \)
23 \( 1 + 4.72iT - 23T^{2} \)
29 \( 1 + 3.13T + 29T^{2} \)
31 \( 1 - 3.52T + 31T^{2} \)
37 \( 1 - 7.10iT - 37T^{2} \)
41 \( 1 - 2.05T + 41T^{2} \)
43 \( 1 + 1.23iT - 43T^{2} \)
47 \( 1 + 10.2iT - 47T^{2} \)
53 \( 1 - 1.98iT - 53T^{2} \)
59 \( 1 - 3.82T + 59T^{2} \)
61 \( 1 + 4.66T + 61T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 - 3.65iT - 73T^{2} \)
79 \( 1 + 4.89T + 79T^{2} \)
83 \( 1 - 0.492iT - 83T^{2} \)
89 \( 1 - 5.50T + 89T^{2} \)
97 \( 1 + 8.86iT - 97T^{2} \)
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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

−8.393272726907281224898253490344, −7.68633913679068000066865522708, −7.05536359544228495008662238139, −6.35581379847845554253673090839, −5.34219574859543379641784635692, −4.58605806134690876503503660716, −3.89609592819603553036106277139, −2.80744868701169630337050745050, −1.91587339672871949520775624177, −0.57517121842081134036571364140, 0.47390205877290420120528439536, 2.41758260061344086738769362465, 3.01152574852177701260157542208, 3.82782355009107892647032568337, 4.79130573659309244938577440664, 5.40462793702230526698179401755, 6.11161774280310904269581564235, 7.34376708439514382572813884444, 7.75859034963903002796454303178, 8.341164579303965886037010887443

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