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Properties

Label 2-3780-140.39-c0-0-0
Degree $2$
Conductor $3780$
Sign $0.0633 - 0.997i$
Analytic cond. $1.88646$
Root an. cond. $1.37348$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)10-s + (−1.5 − 0.866i)11-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + 0.999i·20-s + 1.73·22-s + (0.499 − 0.866i)25-s + (−0.866 + 0.499i)28-s + (0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)10-s + (−1.5 − 0.866i)11-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + 0.999i·20-s + 1.73·22-s + (0.499 − 0.866i)25-s + (−0.866 + 0.499i)28-s + (0.866 + 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3780\)    =    \(2^{2} \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(1.88646\)
Root analytic conductor: \(1.37348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3780} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3780,\ (\ :0),\ 0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4016874670\)
\(L(\frac12)\) \(\approx\) \(0.4016874670\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
good11 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.452867416679943812962085118794, −8.265572684842395067738091932903, −7.31959561987083383497574883807, −7.00674239876047050829488746153, −5.94712269501634638953711873066, −5.47678939340710753333920015551, −4.23350817576148480131725700762, −3.23075721547655870092607067864, −2.57641450604431871076680021214, −0.862539153404391537261649308146, 0.41098045718325021746797511131, 1.96200740031014242666943260352, 2.93423148756418766396870405859, 3.54331685263521978053454504352, 4.67095094111730527442131036142, 5.41487929657018909889495988891, 6.55209827403635740928914856929, 7.35420327690126036295885916614, 7.888672716710827799269789537496, 8.439927394034295587710206590309

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