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Properties

Label 2-1890-63.38-c1-0-13
Degree $2$
Conductor $1890$
Sign $0.0429 - 0.999i$
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 + (−0.323 + 2.62i)7-s − i·8-s + (0.866 + 0.5i)10-s + (−0.664 + 0.383i)11-s + (3.78 − 2.18i)13-s + (−2.62 − 0.323i)14-s + 16-s + (−1.15 + 1.99i)17-s + (4.19 − 2.42i)19-s + (−0.5 + 0.866i)20-s + (−0.383 − 0.664i)22-s + (−4.84 − 2.79i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.223 − 0.387i)5-s + (−0.122 + 0.992i)7-s − 0.353i·8-s + (0.273 + 0.158i)10-s + (−0.200 + 0.115i)11-s + (1.04 − 0.605i)13-s + (−0.701 − 0.0863i)14-s + 0.250·16-s + (−0.279 + 0.483i)17-s + (0.961 − 0.555i)19-s + (−0.111 + 0.193i)20-s + (−0.0818 − 0.141i)22-s + (−1.01 − 0.583i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0429 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0429 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.0429 - 0.999i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.0429 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.681148784\)
\(L(\frac12)\) \(\approx\) \(1.681148784\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.323 - 2.62i)T \)
good11 \( 1 + (0.664 - 0.383i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.78 + 2.18i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.15 - 1.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.19 + 2.42i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.84 + 2.79i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-9.12 - 5.26i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.05iT - 31T^{2} \)
37 \( 1 + (-4.15 - 7.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.65 + 4.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.94 + 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.46T + 47T^{2} \)
53 \( 1 + (-6.48 - 3.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 9.49iT - 61T^{2} \)
67 \( 1 - 13.7T + 67T^{2} \)
71 \( 1 - 7.50iT - 71T^{2} \)
73 \( 1 + (-10.1 - 5.84i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.11T + 79T^{2} \)
83 \( 1 + (0.254 - 0.440i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.80 - 4.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.61 - 4.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.085436476455126177185537262321, −8.570178523082055105077171256082, −8.078022755440853174290208337057, −6.87682962732623375383438104433, −6.22833750779428262887976087481, −5.42915301367464930155041607467, −4.83772197864127846364282916494, −3.62085064647432747761078871041, −2.58707575790385574348017977057, −1.12431297250345025186483086677, 0.75038031394243688779755783076, 1.94814879016292280840744714564, 3.12032561369219006843374379112, 3.92516258920879420133365828038, 4.66635823528045758562873253205, 5.94799807151929847553450758065, 6.50724447690577020467307665314, 7.70918219143052535832107929392, 8.106590592339031946988955067616, 9.468383355633265132166050884874

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