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Properties

Label 1-12e2-144.43-r1-0-0
Degree $1$
Conductor $144$
Sign $0.0436 + 0.999i$
Analytic cond. $15.4749$
Root an. cond. $15.4749$
Motivic weight $0$
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)5-s + (−0.5 − 0.866i)7-s + (−0.866 + 0.5i)11-s + (0.866 + 0.5i)13-s + 17-s + i·19-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.5i)29-s + (0.5 − 0.866i)31-s + i·35-s + i·37-s + (0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.5 + 0.866i)47-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)5-s + (−0.5 − 0.866i)7-s + (−0.866 + 0.5i)11-s + (0.866 + 0.5i)13-s + 17-s + i·19-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.5i)29-s + (0.5 − 0.866i)31-s + i·35-s + i·37-s + (0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.5 + 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0436 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0436 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.0436 + 0.999i$
Analytic conductor: \(15.4749\)
Root analytic conductor: \(15.4749\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 144,\ (1:\ ),\ 0.0436 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5743380563 + 0.5498090786i\)
\(L(\frac12)\) \(\approx\) \(0.5743380563 + 0.5498090786i\)
\(L(1)\) \(\approx\) \(0.7877104919 + 0.03582358157i\)
\(L(1)\) \(\approx\) \(0.7877104919 + 0.03582358157i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + T \)
19 \( 1 + iT \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + iT \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.991500895139861513227320737880, −26.66601815273114271082089746688, −25.97068831411741657454539176849, −24.87528223295061166007823790197, −23.674233250465601144276955587173, −22.91657202295161739482807435009, −21.91920410825858858464299484111, −20.86453003490367646055235932088, −19.62620494810836198292070706408, −18.717889836962851595552964611, −18.087587146489679809489079908396, −16.3150466222669521591797549310, −15.68177205071274008594829723986, −14.75339726101746512205761966172, −13.34646073611925429800526589951, −12.30019562197926749196715063527, −11.23549865218454174888888290851, −10.22856958870338616741378162669, −8.72076893906401049475030408428, −7.83017896151969732358876330625, −6.451919549860733777205751043670, −5.29979590322912578161242562532, −3.60448428218258480213457913832, −2.64426057259916655358246390336, −0.33641382363030370406326225409, 1.2821233292050656239997456614, 3.390582045258843683156852850021, 4.31801115780409677079080742977, 5.78925414666519558514356720148, 7.34346661170252840214891442537, 8.07415804528588798457056702915, 9.54976878571660080404107218234, 10.625891270472789522859356005723, 11.84298507234836109660290540326, 12.86671427584997149547100420166, 13.84993149271397207148837004293, 15.24470042735742182779540443472, 16.21727922262023642138105524253, 16.89994755656865110644120051458, 18.418436021841968251726096429633, 19.27991340557033528614445334183, 20.43051411623361775706023664726, 20.95831463780039144713703300723, 22.65075774777820486777684227957, 23.41140963734817857755779723667, 23.991748598004533441861678402938, 25.53052670133337688705978172084, 26.22899574814393441009201198644, 27.37829134872430791395091425555, 28.172096120210068423405383048340

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