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Properties

Label 2-1890-7.4-c1-0-24
Degree $2$
Conductor $1890$
Sign $0.991 - 0.126i$
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 + (−0.5 − 0.866i)5-s + (−2 + 1.73i)7-s − 0.999·8-s + (0.499 − 0.866i)10-s + (3 − 5.19i)11-s − 13-s + (−2.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + (0.5 + 0.866i)19-s + 0.999·20-s + 6·22-s + (−0.499 + 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.755 + 0.654i)7-s − 0.353·8-s + (0.158 − 0.273i)10-s + (0.904 − 1.56i)11-s − 0.277·13-s + (−0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + (0.114 + 0.198i)19-s + 0.223·20-s + 1.27·22-s + (−0.0999 + 0.173i)25-s + (−0.0980 − 0.169i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.703740505\)
\(L(\frac12)\) \(\approx\) \(1.703740505\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2 - 1.73i)T \)
good11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.113262008298208550164337655982, −8.395652295001950061533936658740, −7.79634836709399735332245836789, −6.47621320176179525474010816419, −6.20957142800846364434791989696, −5.38698004046889658159198211583, −4.25451118294818722258311174174, −3.54030063847414166810468635270, −2.51970027795881457055930244023, −0.69547852975951092178045813050, 1.03508337550623047099796209779, 2.40101236546992600292112777508, 3.25737791175871312691840904662, 4.31943545370376719900143826786, 4.72310045617574034443565767125, 6.11476285729282930198953442809, 6.95275512190948266848463383833, 7.27821835284747853497581570139, 8.608250286531635030873020527107, 9.533271347907720820280755656424

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