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Properties

Label 2-1350-45.2-c1-0-3
Degree $2$
Conductor $1350$
Sign $-0.843 - 0.537i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (1.22 − 0.328i)7-s + (−0.707 − 0.707i)8-s + (−3 − 1.73i)11-s + (−1.22 − 0.328i)13-s + (0.633 + 1.09i)14-s + (0.500 − 0.866i)16-s + 7.19i·19-s + (0.896 − 3.34i)22-s + (−2.12 + 7.91i)23-s − 1.26i·26-s + (−0.896 + 0.896i)28-s + (−3.63 + 6.29i)29-s + (5.09 + 8.83i)31-s + (0.965 + 0.258i)32-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.462 − 0.124i)7-s + (−0.249 − 0.249i)8-s + (−0.904 − 0.522i)11-s + (−0.339 − 0.0910i)13-s + (0.169 + 0.293i)14-s + (0.125 − 0.216i)16-s + 1.65i·19-s + (0.191 − 0.713i)22-s + (−0.442 + 1.65i)23-s − 0.248i·26-s + (−0.169 + 0.169i)28-s + (−0.674 + 1.16i)29-s + (0.915 + 1.58i)31-s + (0.170 + 0.0457i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.843 - 0.537i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.843 - 0.537i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.157531344\)
\(L(\frac12)\) \(\approx\) \(1.157531344\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1.22 + 0.328i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.22 + 0.328i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 7.19iT - 19T^{2} \)
23 \( 1 + (2.12 - 7.91i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.63 - 6.29i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.09 - 8.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.55 - 1.55i)T + 37iT^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.67 + 6.24i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.55 + 5.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.55 - 1.55i)T + 53iT^{2} \)
59 \( 1 + (-6.23 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.22 + 12.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + (3.67 - 3.67i)T - 73iT^{2} \)
79 \( 1 + (8.66 + 5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.57 + 1.76i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 8.66T + 89T^{2} \)
97 \( 1 + (14.1 - 3.79i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.986643608093022412905709140278, −8.911102526463198614521680284317, −8.123257999844449683152469873532, −7.61097213477567870630612404853, −6.71506031695697831115687761922, −5.54319668414727942368793077668, −5.26238239017561113658206902793, −3.98159007773357854745308448620, −3.10317787777367618513235580995, −1.54937467047422783724815698773, 0.44154318442811430177438717557, 2.19813725047283572089435331406, 2.73480198430516905028304774726, 4.32013831541123853325669470870, 4.73426680912911011030958498197, 5.79127283054674800649735655105, 6.79737794719785438040461476678, 7.84994238946129118784052682427, 8.446891766069186819411867477675, 9.581686818763033775308563356647

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