Properties

Label 2-1890-63.5-c1-0-27
Degree $2$
Conductor $1890$
Sign $0.165 + 0.986i$
Analytic cond. $15.0917$
Root an. cond. $3.88480$
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·2-s − 4-s + (0.5 + 0.866i)5-s + (−2.21 − 1.44i)7-s i·8-s + (−0.866 + 0.5i)10-s + (4.20 + 2.43i)11-s + (−3.42 − 1.97i)13-s + (1.44 − 2.21i)14-s + 16-s + (−1.25 − 2.16i)17-s + (−0.962 − 0.555i)19-s + (−0.5 − 0.866i)20-s + (−2.43 + 4.20i)22-s + (−2.86 + 1.65i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.223 + 0.387i)5-s + (−0.837 − 0.545i)7-s − 0.353i·8-s + (−0.273 + 0.158i)10-s + (1.26 + 0.732i)11-s + (−0.948 − 0.547i)13-s + (0.385 − 0.592i)14-s + 0.250·16-s + (−0.303 − 0.526i)17-s + (−0.220 − 0.127i)19-s + (−0.111 − 0.193i)20-s + (−0.518 + 0.897i)22-s + (−0.596 + 0.344i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.165 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5826337411\)
\(L(\frac12)\) \(\approx\) \(0.5826337411\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.21 + 1.44i)T \)
good11 \( 1 + (-4.20 - 2.43i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.42 + 1.97i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.25 + 2.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.962 + 0.555i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.86 - 1.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.70 - 2.14i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.39iT - 31T^{2} \)
37 \( 1 + (1.82 - 3.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.90 + 6.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.62 + 6.28i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + (4.92 - 2.84i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.39T + 59T^{2} \)
61 \( 1 + 9.20iT - 61T^{2} \)
67 \( 1 - 6.01T + 67T^{2} \)
71 \( 1 + 9.66iT - 71T^{2} \)
73 \( 1 + (-12.0 + 6.96i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 7.83T + 79T^{2} \)
83 \( 1 + (0.393 + 0.682i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.49 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.9 - 8.64i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.358864260714313876955320342679, −8.058624701394824012686665961721, −7.25250676229432420562939836724, −6.77830624057246255238080766395, −6.06790180891004382194882287517, −5.04734918370174605421433180269, −4.11753645209378471503485366762, −3.30143871955816737246886106385, −1.97003939404294085599550204457, −0.21139480347774343066064230110, 1.40344639227553989099431378192, 2.45737904811221937463845255291, 3.50362622867779669391368146066, 4.28809842641865395893631975581, 5.30174153472750380372874511937, 6.25465564757951246096085640283, 6.78794628338020007231006235761, 8.181603337967227911676083755073, 8.779522340202016124648384325922, 9.577514006771077263229991896003

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