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Properties

Label 2-4020-201.200-c1-0-73
Degree $2$
Conductor $4020$
Sign $0.154 + 0.988i$
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  + (1.70 − 0.322i)3-s − 5-s − 1.22i·7-s + (2.79 − 1.09i)9-s − 1.70·11-s − 4.75i·13-s + (−1.70 + 0.322i)15-s + 6.23i·17-s + 7.09·19-s + (−0.396 − 2.09i)21-s − 1.34i·23-s + 25-s + (4.39 − 2.76i)27-s − 2.58i·29-s − 5.45i·31-s + ⋯
L(s)  = 1  + (0.982 − 0.186i)3-s − 0.447·5-s − 0.464i·7-s + (0.930 − 0.366i)9-s − 0.513·11-s − 1.31i·13-s + (−0.439 + 0.0833i)15-s + 1.51i·17-s + 1.62·19-s + (−0.0865 − 0.456i)21-s − 0.280i·23-s + 0.200·25-s + (0.846 − 0.533i)27-s − 0.480i·29-s − 0.980i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.154 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.154 + 0.988i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.154 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.369793718\)
\(L(\frac12)\) \(\approx\) \(2.369793718\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.322i)T \)
5 \( 1 + T \)
67 \( 1 + (0.267 - 8.18i)T \)
good7 \( 1 + 1.22iT - 7T^{2} \)
11 \( 1 + 1.70T + 11T^{2} \)
13 \( 1 + 4.75iT - 13T^{2} \)
17 \( 1 - 6.23iT - 17T^{2} \)
19 \( 1 - 7.09T + 19T^{2} \)
23 \( 1 + 1.34iT - 23T^{2} \)
29 \( 1 + 2.58iT - 29T^{2} \)
31 \( 1 + 5.45iT - 31T^{2} \)
37 \( 1 + 6.93T + 37T^{2} \)
41 \( 1 + 3.76T + 41T^{2} \)
43 \( 1 + 1.89iT - 43T^{2} \)
47 \( 1 + 0.286iT - 47T^{2} \)
53 \( 1 - 3.48T + 53T^{2} \)
59 \( 1 + 0.962iT - 59T^{2} \)
61 \( 1 + 1.51iT - 61T^{2} \)
71 \( 1 + 6.96iT - 71T^{2} \)
73 \( 1 - 0.332T + 73T^{2} \)
79 \( 1 + 8.41iT - 79T^{2} \)
83 \( 1 + 9.17iT - 83T^{2} \)
89 \( 1 + 4.45iT - 89T^{2} \)
97 \( 1 + 9.67iT - 97T^{2} \)
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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

−8.237206930492796284080783721866, −7.60007090673743592647046768116, −7.19101173265328971566000604613, −6.07521531899389275599496878788, −5.28845749181492968006502334319, −4.25225058383399864385445052508, −3.51781942616023840742229095909, −2.92639084827358378886096409011, −1.78277386279317774095972509496, −0.62616132133911557899918376860, 1.27071849477799403521786544993, 2.39971147105008305849230671780, 3.12901626643055752083968910206, 3.86899849809090410042921973071, 4.93478591422245337866668733619, 5.29461835099059142631807289210, 6.79138652422892403054462686439, 7.19022866775375854766848772004, 7.88011680894762120839815638422, 8.742900186397633225482466848917

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