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Properties

Label 2-1890-63.25-c1-0-21
Degree $2$
Conductor $1890$
Sign $-0.215 + 0.976i$
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  − 2-s + 4-s + (0.5 − 0.866i)5-s + (1.40 − 2.24i)7-s − 8-s + (−0.5 + 0.866i)10-s + (0.357 + 0.618i)11-s + (−0.823 − 1.42i)13-s + (−1.40 + 2.24i)14-s + 16-s + (1.61 − 2.79i)17-s + (−0.0769 − 0.133i)19-s + (0.5 − 0.866i)20-s + (−0.357 − 0.618i)22-s + (1.43 − 2.49i)23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.531 − 0.847i)7-s − 0.353·8-s + (−0.158 + 0.273i)10-s + (0.107 + 0.186i)11-s + (−0.228 − 0.395i)13-s + (−0.375 + 0.599i)14-s + 0.250·16-s + (0.391 − 0.678i)17-s + (−0.0176 − 0.0305i)19-s + (0.111 − 0.193i)20-s + (−0.0761 − 0.131i)22-s + (0.300 − 0.519i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.170309371\)
\(L(\frac12)\) \(\approx\) \(1.170309371\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.40 + 2.24i)T \)
good11 \( 1 + (-0.357 - 0.618i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.823 + 1.42i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.61 + 2.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0769 + 0.133i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.43 + 2.49i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.602 + 1.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.40T + 31T^{2} \)
37 \( 1 + (-4.69 - 8.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.09 + 5.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.80 + 8.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.29T + 47T^{2} \)
53 \( 1 + (-0.576 + 0.999i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.24T + 59T^{2} \)
61 \( 1 + 9.20T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 3.71T + 71T^{2} \)
73 \( 1 + (1.45 - 2.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.44T + 79T^{2} \)
83 \( 1 + (2.21 - 3.83i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.52 - 9.57i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.23 - 7.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.011218736138418191500703513329, −8.239771995870323581085688244396, −7.46220810276871735090901402769, −6.94299522718748324692705557907, −5.80565144376558296615171378107, −4.96557456304630948338462267449, −4.04760769826333482453667143135, −2.84161108169208927416496763245, −1.64662775806904410195163586610, −0.56113994497282297186304591507, 1.42405107629118300186058390271, 2.35285548446078599984918770520, 3.36138502023465266084548611885, 4.59484173676331256630717792967, 5.72762559756290079613550507804, 6.17768005026363095898424924263, 7.34545095032843999389039164384, 7.83173990311677045157251479626, 8.880797658283269296134725110999, 9.232645680278440480702860554990

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