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Properties

Label 2-3780-1260.1139-c0-0-2
Degree $2$
Conductor $3780$
Sign $-0.0477 - 0.998i$
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  + i·2-s − 4-s + (0.5 + 0.866i)5-s + (0.866 − 0.5i)7-s − i·8-s + (−0.866 + 0.5i)10-s + (0.5 + 0.866i)14-s + 16-s + (−0.5 − 0.866i)20-s + (1.73 − i)23-s + (−0.499 + 0.866i)25-s + (−0.866 + 0.5i)28-s + (−1.5 + 0.866i)29-s + i·32-s + (0.866 + 0.499i)35-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (0.5 + 0.866i)5-s + (0.866 − 0.5i)7-s − i·8-s + (−0.866 + 0.5i)10-s + (0.5 + 0.866i)14-s + 16-s + (−0.5 − 0.866i)20-s + (1.73 − i)23-s + (−0.499 + 0.866i)25-s + (−0.866 + 0.5i)28-s + (−1.5 + 0.866i)29-s + i·32-s + (0.866 + 0.499i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3780\)    =    \(2^{2} \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.0477 - 0.998i$
Analytic conductor: \(1.88646\)
Root analytic conductor: \(1.37348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3780} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3780,\ (\ :0),\ -0.0477 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.433150492\)
\(L(\frac12)\) \(\approx\) \(1.433150492\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.905996907780224501003919339868, −7.891964129747647026928565232251, −7.27403706673030112607240113725, −6.85902356038518733417331691829, −5.92397771295446016858480701558, −5.28448092532479341020117509292, −4.47628756609211760864500103869, −3.62836173022259534221037255291, −2.55750885766221315063554289998, −1.20729846941994540173750911848, 1.04523359969477629893059216548, 1.90822940096904131168005998027, 2.72486050753318289266964307112, 3.92161153970244716294860416340, 4.61647695367494485258921257699, 5.50508433053547187506251790600, 5.69813883189996371958727996050, 7.26046809940268351173470701375, 7.987942332971501487530287620045, 8.828751490627442229885608665397

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