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Properties

Label 2-1350-45.23-c1-0-0
Degree $2$
Conductor $1350$
Sign $-0.999 + 0.0438i$
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 + (0.707 − 0.707i)8-s + (1.5 − 0.866i)11-s + (−6.69 + 1.79i)13-s + (0.500 + 0.866i)16-s + (2.12 + 2.12i)17-s + 4i·19-s + (0.448 + 1.67i)22-s + (−1.55 − 5.79i)23-s − 6.92i·26-s + (−1.73 − 3i)29-s + (−2 + 3.46i)31-s + (−0.965 + 0.258i)32-s + (−2.59 + 1.5i)34-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.433 − 0.249i)4-s + (0.249 − 0.249i)8-s + (0.452 − 0.261i)11-s + (−1.85 + 0.497i)13-s + (0.125 + 0.216i)16-s + (0.514 + 0.514i)17-s + 0.917i·19-s + (0.0955 + 0.356i)22-s + (−0.323 − 1.20i)23-s − 1.35i·26-s + (−0.321 − 0.557i)29-s + (−0.359 + 0.622i)31-s + (−0.170 + 0.0457i)32-s + (−0.445 + 0.257i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.999 + 0.0438i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.999 + 0.0438i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5181685486\)
\(L(\frac12)\) \(\approx\) \(0.5181685486\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.69 - 1.79i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.12 - 2.12i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (1.55 + 5.79i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.73 + 3i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 37iT^{2} \)
41 \( 1 + (-6 - 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.24 - 8.36i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.10 - 11.5i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (8.48 - 8.48i)T - 53iT^{2} \)
59 \( 1 + (-4.33 + 7.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.896 + 3.34i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (4.89 + 4.89i)T + 73iT^{2} \)
79 \( 1 + (3.46 - 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.69 + 2.32i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 1.73T + 89T^{2} \)
97 \( 1 + (-15.0 - 4.03i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.779753977732106230128412003664, −9.299640601790853607931629416498, −8.155649833607222499069806868489, −7.72599988288043769268631978977, −6.67167972423168670594132136613, −6.09598159153714513931529611466, −4.97729115342234661541062368047, −4.29743085492663453619508895249, −3.01375762139777929502866271207, −1.61649557509912074052416837839, 0.21951575038598224025091763543, 1.84983635150204125305458666923, 2.84407914933713205183865672503, 3.85353312763689620130278964377, 4.95785311268417168538749682605, 5.56451578728606913533872093224, 7.17023201864889028445796798228, 7.38476301247209135402788528436, 8.581104677467294403482427175646, 9.479622483780203901080683513994

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