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Properties

Label 2-1890-63.5-c1-0-10
Degree $2$
Conductor $1890$
Sign $0.904 - 0.427i$
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 + (1.55 + 2.13i)7-s + i·8-s + (0.866 − 0.5i)10-s + (3.90 + 2.25i)11-s + (−1.92 − 1.11i)13-s + (2.13 − 1.55i)14-s + 16-s + (−3.31 − 5.74i)17-s + (6.71 + 3.87i)19-s + (−0.5 − 0.866i)20-s + (2.25 − 3.90i)22-s + (−4.34 + 2.51i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.223 + 0.387i)5-s + (0.588 + 0.808i)7-s + 0.353i·8-s + (0.273 − 0.158i)10-s + (1.17 + 0.680i)11-s + (−0.534 − 0.308i)13-s + (0.571 − 0.416i)14-s + 0.250·16-s + (−0.803 − 1.39i)17-s + (1.53 + 0.888i)19-s + (−0.111 − 0.193i)20-s + (0.480 − 0.832i)22-s + (−0.906 + 0.523i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.904 - 0.427i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.904 - 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.784472699\)
\(L(\frac12)\) \(\approx\) \(1.784472699\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 - 2.13i)T \)
good11 \( 1 + (-3.90 - 2.25i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.92 + 1.11i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.31 + 5.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.71 - 3.87i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.34 - 2.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.08 - 0.623i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.99iT - 31T^{2} \)
37 \( 1 + (0.475 - 0.822i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.31 + 7.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.87 - 8.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.06T + 47T^{2} \)
53 \( 1 + (4.68 - 2.70i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.27T + 59T^{2} \)
61 \( 1 - 11.4iT - 61T^{2} \)
67 \( 1 - 5.53T + 67T^{2} \)
71 \( 1 - 9.57iT - 71T^{2} \)
73 \( 1 + (-1.11 + 0.642i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 3.96T + 79T^{2} \)
83 \( 1 + (-5.71 - 9.90i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.89 + 15.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.24 + 4.18i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.555829250184657655040839148828, −8.736661448361191726704472164892, −7.68834337652121335575687812402, −7.04836817216214280637288690951, −5.89912014409568559794328911082, −5.15510466335368321537637112471, −4.29997347467593980013045543841, −3.20166364357797679602970069841, −2.29929973598124546472151618736, −1.34441503186174735061223005554, 0.71875459394548203397654172336, 1.93211667788885855949287438221, 3.62571325379199063149293074293, 4.31252653928257656561263698482, 5.10570028708124928189567177818, 6.14901943694861765526368030839, 6.69481934054873646092085511474, 7.67186945489917679633057463998, 8.249632789969274027121794264485, 9.149275433866624961534107789405

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