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Properties

Label 2-1890-315.209-c1-0-2
Degree $2$
Conductor $1890$
Sign $-0.128 - 0.991i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.12 − 0.703i)5-s + (−2.48 − 0.896i)7-s + 0.999·8-s + (1.67 − 1.48i)10-s + (−5.48 − 3.16i)11-s + (−2.28 − 3.95i)13-s + (2.02 − 1.70i)14-s + (−0.5 + 0.866i)16-s + 2.72i·17-s − 4.11i·19-s + (0.452 + 2.18i)20-s + (5.48 − 3.16i)22-s + (1.15 + 2.00i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.949 − 0.314i)5-s + (−0.940 − 0.339i)7-s + 0.353·8-s + (0.528 − 0.470i)10-s + (−1.65 − 0.953i)11-s + (−0.632 − 1.09i)13-s + (0.540 − 0.456i)14-s + (−0.125 + 0.216i)16-s + 0.661i·17-s − 0.944i·19-s + (0.101 + 0.489i)20-s + (1.16 − 0.674i)22-s + (0.241 + 0.417i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.128 - 0.991i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.128 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2655955349\)
\(L(\frac12)\) \(\approx\) \(0.2655955349\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (2.12 + 0.703i)T \)
7 \( 1 + (2.48 + 0.896i)T \)
good11 \( 1 + (5.48 + 3.16i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.28 + 3.95i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.72iT - 17T^{2} \)
19 \( 1 + 4.11iT - 19T^{2} \)
23 \( 1 + (-1.15 - 2.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.173 - 0.100i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.10 - 1.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.4iT - 37T^{2} \)
41 \( 1 + (1.03 + 1.78i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.26 - 4.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.86 + 4.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.77T + 53T^{2} \)
59 \( 1 + (0.616 + 1.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.66 + 0.961i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.07 + 4.08i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.31iT - 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 + (-1.01 + 1.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.64 + 2.10i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.48T + 89T^{2} \)
97 \( 1 + (-2.42 + 4.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.273830036323952293458621138265, −8.436954493285923081799013803554, −7.86193509145558191366995314009, −7.29422500410745468885923010002, −6.34158012095714735218699329759, −5.40162918091215814698450164131, −4.78100992494603693661034136752, −3.47966303744153244748386879072, −2.81630050122587956797943783538, −0.69691827494442971505556034164, 0.17258982246993538379289732369, 2.20466026123493014347750829157, 2.83475016358853151636695821827, 3.91504527567415651108661609872, 4.69781325151391105285866527407, 5.73571617770220007469576501245, 7.10096287573562273212224282657, 7.32317461127181614703413903002, 8.265882665870098407405627295753, 9.151880192635569602935035286000

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