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Properties

Label 2-1890-63.38-c1-0-16
Degree $2$
Conductor $1890$
Sign $0.822 + 0.568i$
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 + (0.145 + 2.64i)7-s + i·8-s + (−0.866 − 0.5i)10-s + (2.74 − 1.58i)11-s + (1.82 − 1.05i)13-s + (2.64 − 0.145i)14-s + 16-s + (−0.0900 + 0.155i)17-s + (−5.17 + 2.98i)19-s + (−0.5 + 0.866i)20-s + (−1.58 − 2.74i)22-s + (0.683 + 0.394i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.223 − 0.387i)5-s + (0.0549 + 0.998i)7-s + 0.353i·8-s + (−0.273 − 0.158i)10-s + (0.828 − 0.478i)11-s + (0.505 − 0.292i)13-s + (0.706 − 0.0388i)14-s + 0.250·16-s + (−0.0218 + 0.0378i)17-s + (−1.18 + 0.685i)19-s + (−0.111 + 0.193i)20-s + (−0.338 − 0.586i)22-s + (0.142 + 0.0822i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.822 + 0.568i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.822 + 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.814145682\)
\(L(\frac12)\) \(\approx\) \(1.814145682\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.145 - 2.64i)T \)
good11 \( 1 + (-2.74 + 1.58i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.82 + 1.05i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.0900 - 0.155i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.17 - 2.98i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.683 - 0.394i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.84 - 3.95i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.70iT - 31T^{2} \)
37 \( 1 + (-4.27 - 7.39i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.85 - 10.1i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.84 - 3.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.08T + 47T^{2} \)
53 \( 1 + (-0.613 - 0.353i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 2.89iT - 61T^{2} \)
67 \( 1 - 9.95T + 67T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (5.09 + 2.93i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + (-4.41 + 7.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.91 - 5.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.79 - 2.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.223194134850311892250873566435, −8.426795885895892179985456410724, −8.046553799382043252445281659113, −6.37133107142634883742004015871, −6.05526642155056617569836347193, −4.95856680331593948911121401974, −4.12840723372322662454929207971, −3.10559961761040268103658395976, −2.12185663101969420366110031486, −1.02210676410407520656042626572, 0.885084934282038381705988909625, 2.31822629850114996049804580964, 3.79603452403501856573078152296, 4.28634254239736495154743160236, 5.31261059083275774917026070998, 6.50698320155183738632783941737, 6.75424183701087599447089014061, 7.52975049722325652906560724872, 8.562619874334297903714817901630, 9.093118462054476969746199774451

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