Properties

Degree $2$
Conductor $144$
Sign $0.380 - 0.924i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.957 + 1.04i)2-s + (−0.167 + 1.99i)4-s + (−0.236 − 0.236i)5-s + 3.27·7-s + (−2.23 + 1.73i)8-s + (0.0197 − 0.472i)10-s + (−2.58 + 2.58i)11-s + (−1.70 − 1.70i)13-s + (3.13 + 3.41i)14-s + (−3.94 − 0.665i)16-s − 7.05i·17-s + (3.04 − 3.04i)19-s + (0.510 − 0.431i)20-s + (−5.16 − 0.215i)22-s + 1.47i·23-s + ⋯
L(s)  = 1  + (0.676 + 0.736i)2-s + (−0.0835 + 0.996i)4-s + (−0.105 − 0.105i)5-s + 1.23·7-s + (−0.790 + 0.613i)8-s + (0.00624 − 0.149i)10-s + (−0.778 + 0.778i)11-s + (−0.473 − 0.473i)13-s + (0.838 + 0.912i)14-s + (−0.986 − 0.166i)16-s − 1.71i·17-s + (0.697 − 0.697i)19-s + (0.114 − 0.0964i)20-s + (−1.10 − 0.0460i)22-s + 0.307i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.380 - 0.924i$
Motivic weight: \(1\)
Character: $\chi_{144} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.380 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27315 + 0.853150i\)
\(L(\frac12)\) \(\approx\) \(1.27315 + 0.853150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.957 - 1.04i)T \)
3 \( 1 \)
good5 \( 1 + (0.236 + 0.236i)T + 5iT^{2} \)
7 \( 1 - 3.27T + 7T^{2} \)
11 \( 1 + (2.58 - 2.58i)T - 11iT^{2} \)
13 \( 1 + (1.70 + 1.70i)T + 13iT^{2} \)
17 \( 1 + 7.05iT - 17T^{2} \)
19 \( 1 + (-3.04 + 3.04i)T - 19iT^{2} \)
23 \( 1 - 1.47iT - 23T^{2} \)
29 \( 1 + (-2.98 + 2.98i)T - 29iT^{2} \)
31 \( 1 - 8.02iT - 31T^{2} \)
37 \( 1 + (7.93 - 7.93i)T - 37iT^{2} \)
41 \( 1 + 2.22T + 41T^{2} \)
43 \( 1 + (4.61 + 4.61i)T + 43iT^{2} \)
47 \( 1 - 7.13T + 47T^{2} \)
53 \( 1 + (-5.81 - 5.81i)T + 53iT^{2} \)
59 \( 1 + (7.46 - 7.46i)T - 59iT^{2} \)
61 \( 1 + (4.04 + 4.04i)T + 61iT^{2} \)
67 \( 1 + (2.90 - 2.90i)T - 67iT^{2} \)
71 \( 1 - 1.02iT - 71T^{2} \)
73 \( 1 - 4.08iT - 73T^{2} \)
79 \( 1 - 5.36iT - 79T^{2} \)
83 \( 1 + (-3.93 - 3.93i)T + 83iT^{2} \)
89 \( 1 - 2.35T + 89T^{2} \)
97 \( 1 - 9.97T + 97T^{2} \)
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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

−13.63140147902841204418787937670, −12.23327521384256675837035554751, −11.70634735875621770728144731500, −10.28156057798316753879021431971, −8.810456573474953707151757141888, −7.74767789827472264456886940307, −7.02754648881615774295247761092, −5.16644950029286219589417011985, −4.76956211397914526289559941126, −2.75380374741549645899048178440, 1.85147493123057283421607396291, 3.61173170693009651097098897032, 4.94977054773193551927573861402, 5.96596832280954839006870747244, 7.67037636530522169022591297091, 8.811385730400780684053251877673, 10.27951933886917884359370622990, 10.99740056492944220445035349569, 11.85811401551855652882129836960, 12.85780460087394652479171934846

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