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Properties

Label 2-1350-45.34-c1-0-7
Degree $2$
Conductor $1350$
Sign $0.993 + 0.114i$
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 + (1.73 + i)7-s − 0.999i·8-s + (−3.46 + 2i)13-s + (−0.999 − 1.73i)14-s + (−0.5 + 0.866i)16-s − 6i·17-s + 7·19-s + 3.99·26-s + 1.99i·28-s + (3 − 5.19i)29-s + (5 + 8.66i)31-s + (0.866 − 0.499i)32-s + (−3 + 5.19i)34-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.654 + 0.377i)7-s − 0.353i·8-s + (−0.960 + 0.554i)13-s + (−0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s − 1.45i·17-s + 1.60·19-s + 0.784·26-s + 0.377i·28-s + (0.557 − 0.964i)29-s + (0.898 + 1.55i)31-s + (0.153 − 0.0883i)32-s + (−0.514 + 0.891i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.993 + 0.114i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.993 + 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.311373949\)
\(L(\frac12)\) \(\approx\) \(1.311373949\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.46 - 2i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.2 + 6.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.79 - 4.5i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (-14.7 - 8.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.562922072966498269575091716291, −8.969018611612823151844443800113, −7.947798439953006710358022092950, −7.39150615089622732904565077969, −6.50810814000097196333523990706, −5.19809772606152722110062486591, −4.62866120432971315878365777365, −3.14702222612980849331976957263, −2.33183285573238647199086779998, −0.987310829189294779896548620128, 0.903271378765730089501787412730, 2.17361502715686166069118336606, 3.49423706772810685778346609235, 4.72260056371643337393715271735, 5.48624118257341965937853219517, 6.44414379649461042427532522868, 7.45094903391604556557285386178, 7.910702448875800531573715918971, 8.681862411762246513796754643199, 9.748432258607938996398990439640

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