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Properties

Label 2-1890-63.38-c1-0-30
Degree $2$
Conductor $1890$
Sign $-0.994 - 0.100i$
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 + (2.55 − 0.696i)7-s + i·8-s + (−0.866 − 0.5i)10-s + (−1.26 + 0.732i)11-s + (−6.03 + 3.48i)13-s + (−0.696 − 2.55i)14-s + 16-s + (3.30 − 5.72i)17-s + (−3.08 + 1.78i)19-s + (−0.5 + 0.866i)20-s + (0.732 + 1.26i)22-s + (−5.51 − 3.18i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.223 − 0.387i)5-s + (0.964 − 0.263i)7-s + 0.353i·8-s + (−0.273 − 0.158i)10-s + (−0.382 + 0.220i)11-s + (−1.67 + 0.966i)13-s + (−0.186 − 0.682i)14-s + 0.250·16-s + (0.801 − 1.38i)17-s + (−0.707 + 0.408i)19-s + (−0.111 + 0.193i)20-s + (0.156 + 0.270i)22-s + (−1.14 − 0.663i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.994 - 0.100i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8392992560\)
\(L(\frac12)\) \(\approx\) \(0.8392992560\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.55 + 0.696i)T \)
good11 \( 1 + (1.26 - 0.732i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.03 - 3.48i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.30 + 5.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.08 - 1.78i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.51 + 3.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.49 + 0.862i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.84iT - 31T^{2} \)
37 \( 1 + (2.75 + 4.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.632 + 1.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.24 - 5.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.39T + 47T^{2} \)
53 \( 1 + (5.60 + 3.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.49T + 59T^{2} \)
61 \( 1 + 3.73iT - 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 14.2iT - 71T^{2} \)
73 \( 1 + (1.11 + 0.640i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.994T + 79T^{2} \)
83 \( 1 + (-5.93 + 10.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.18 + 3.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.04 - 5.22i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.032021067920249759847396636248, −7.909678256039970132659813942851, −7.59891021038635539334346545361, −6.39908681852532120620904781983, −5.14244283718538752855816300363, −4.78137480321861288742362484946, −3.89711936974740717266482593232, −2.42564605309865888085225952534, −1.87656168436328207239921748156, −0.28689152096734883812216896232, 1.66422797015485790415531055629, 2.81921787391119257799246583304, 3.96087138831305291055208054968, 5.16390270456210854864815396888, 5.42329250987648223225749097701, 6.48786165975302315715410670338, 7.35634991393195787817092647015, 8.105916525616277971200997964865, 8.446477759327773582775000762420, 9.693602751097290495595225717237

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