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Properties

Label 2-1350-45.23-c1-0-5
Degree $2$
Conductor $1350$
Sign $0.718 - 0.695i$
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 + (−3 + 1.73i)11-s + (−3.34 + 0.896i)13-s + (0.500 + 0.866i)16-s + (4.24 + 4.24i)17-s − 5i·19-s + (0.896 + 3.34i)22-s + (1.55 + 5.79i)23-s + 3.46i·26-s + (3.46 + 6i)29-s + (−2 + 3.46i)31-s + (0.965 − 0.258i)32-s + (5.19 − 3i)34-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.433 − 0.249i)4-s + (−0.249 + 0.249i)8-s + (−0.904 + 0.522i)11-s + (−0.928 + 0.248i)13-s + (0.125 + 0.216i)16-s + (1.02 + 1.02i)17-s − 1.14i·19-s + (0.191 + 0.713i)22-s + (0.323 + 1.20i)23-s + 0.679i·26-s + (0.643 + 1.11i)29-s + (−0.359 + 0.622i)31-s + (0.170 − 0.0457i)32-s + (0.891 − 0.514i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.718 - 0.695i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.718 - 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.145295894\)
\(L(\frac12)\) \(\approx\) \(1.145295894\)
\(L(\frac{3}{2})\) 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 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.34 - 0.896i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \)
19 \( 1 + 5iT - 19T^{2} \)
23 \( 1 + (-1.55 - 5.79i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.46 - 6i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.13 - 11.7i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.55 - 5.79i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.24 - 4.24i)T - 53iT^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.24 - 8.36i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 + (-8.57 - 8.57i)T + 73iT^{2} \)
79 \( 1 + (-12.1 + 7i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.69 + 2.32i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (5.01 + 1.34i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.724161298438570102853519904826, −9.206891299157294386893327671885, −8.032975358910638329383976204794, −7.42473339274948867704898629537, −6.34285243142654495945121273474, −5.19098925947180194664172497624, −4.73152932044974407122251286194, −3.44301172818004495886731855967, −2.60254083350084254682804552394, −1.39200427565062731752158788716, 0.45574769133996219019952003101, 2.45519470471167613237535167824, 3.41461576166585817468218949179, 4.65338818236856342568072877709, 5.35164366247996133948552801164, 6.13501835132612877079584236457, 7.12804988183107077645717804604, 7.931641160993927717214088851391, 8.349779984078981138539972219867, 9.624630264886225788632339803289

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