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Properties

Label 2-1890-63.5-c1-0-17
Degree $2$
Conductor $1890$
Sign $-0.00672 - 0.999i$
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 + (2.64 + 0.162i)7-s − i·8-s + (−0.866 + 0.5i)10-s + (0.650 + 0.375i)11-s + (2.72 + 1.57i)13-s + (−0.162 + 2.64i)14-s + 16-s + (0.433 + 0.750i)17-s + (1.23 + 0.713i)19-s + (−0.5 − 0.866i)20-s + (−0.375 + 0.650i)22-s + (−4.38 + 2.53i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.223 + 0.387i)5-s + (0.998 + 0.0612i)7-s − 0.353i·8-s + (−0.273 + 0.158i)10-s + (0.196 + 0.113i)11-s + (0.754 + 0.435i)13-s + (−0.0433 + 0.705i)14-s + 0.250·16-s + (0.105 + 0.181i)17-s + (0.283 + 0.163i)19-s + (−0.111 − 0.193i)20-s + (−0.0800 + 0.138i)22-s + (−0.914 + 0.527i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00672 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00672 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.00672 - 0.999i$
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: \((2,\ 1890,\ (\ :1/2),\ -0.00672 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.027514680\)
\(L(\frac12)\) \(\approx\) \(2.027514680\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 0.162i)T \)
good11 \( 1 + (-0.650 - 0.375i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.72 - 1.57i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.433 - 0.750i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.23 - 0.713i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.38 - 2.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.23 + 3.02i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.21iT - 31T^{2} \)
37 \( 1 + (-3.60 + 6.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.08 - 3.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.69 - 6.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.45T + 47T^{2} \)
53 \( 1 + (0.790 - 0.456i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.43T + 59T^{2} \)
61 \( 1 - 10.7iT - 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 - 5.43iT - 71T^{2} \)
73 \( 1 + (-0.539 + 0.311i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.00585T + 79T^{2} \)
83 \( 1 + (-1.01 - 1.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.22 - 2.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.23 - 5.33i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.331306017117335781182616219245, −8.437502098704844654805923593669, −7.85420094906703631571308170285, −7.12330323910133859418586422621, −6.09959030998139467296505146720, −5.67987797358504090217512845265, −4.48899382697104161600717019303, −3.88858536717672162528262326397, −2.45510396556538349980040412252, −1.25240240679031283293756204642, 0.892917201846617512447985123418, 1.82874530630130302306437421339, 2.99690449462661642843199293746, 4.04076727348304649977318929011, 4.86970771628035412642305648067, 5.57543715790137181020616714917, 6.58551683029715831628302171547, 7.70447254288369634212236221128, 8.494507420242429184401250867033, 8.871120765281098463036747472904

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