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Properties

Label 2-1350-45.34-c1-0-17
Degree $2$
Conductor $1350$
Sign $0.397 + 0.917i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−3.46 − 2i)7-s + 0.999i·8-s + (1.5 − 2.59i)11-s + (−3.46 + 2i)13-s + (−1.99 − 3.46i)14-s + (−0.5 + 0.866i)16-s − 3i·17-s + 4·19-s + (2.59 − 1.5i)22-s + (5.19 − 3i)23-s − 3.99·26-s − 3.99i·28-s + (3 − 5.19i)29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.30 − 0.755i)7-s + 0.353i·8-s + (0.452 − 0.783i)11-s + (−0.960 + 0.554i)13-s + (−0.534 − 0.925i)14-s + (−0.125 + 0.216i)16-s − 0.727i·17-s + 0.917·19-s + (0.553 − 0.319i)22-s + (1.08 − 0.625i)23-s − 0.784·26-s − 0.755i·28-s + (0.557 − 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.397 + 0.917i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.397 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.570083639\)
\(L(\frac12)\) \(\approx\) \(1.570083639\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (3.46 + 2i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.46 - 2i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.3 + 6i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.46 + 2i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.79 + 4.5i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (6.06 + 3.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.507274221794895867145966053909, −8.704510711467057856403973119334, −7.40915079925229517464564549703, −7.03306373353913320430697912511, −6.20624551650071824098263319084, −5.30818309224393171443392242219, −4.24731318395897550072025837613, −3.44860247441931919091062349752, −2.54135215374031022065314848866, −0.52001905300213681786678967794, 1.54560008975365690710728840645, 2.96298453664751836794213367225, 3.36099241622914863781522462628, 4.82638874836616435354094745209, 5.38562547495512621390246873111, 6.53627746934021715804196715200, 6.96103248341772791580574854932, 8.150144666234174911310033726472, 9.348439282594727545308613741123, 9.661694263550316341077268300903

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