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Properties

Label 2-12e2-144.13-c1-0-9
Degree $2$
Conductor $144$
Sign $0.869 + 0.493i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + (−1.17 − 0.782i)2-s + (1.34 + 1.08i)3-s + (0.776 + 1.84i)4-s + (−0.777 − 2.90i)5-s + (−0.738 − 2.33i)6-s + (1.04 − 0.603i)7-s + (0.526 − 2.77i)8-s + (0.636 + 2.93i)9-s + (−1.35 + 4.02i)10-s + (5.09 + 1.36i)11-s + (−0.956 + 3.32i)12-s + (2.02 − 0.541i)13-s + (−1.70 − 0.106i)14-s + (2.10 − 4.75i)15-s + (−2.79 + 2.86i)16-s − 3.20·17-s + ⋯
L(s)  = 1  + (−0.833 − 0.553i)2-s + (0.778 + 0.627i)3-s + (0.388 + 0.921i)4-s + (−0.347 − 1.29i)5-s + (−0.301 − 0.953i)6-s + (0.395 − 0.228i)7-s + (0.186 − 0.982i)8-s + (0.212 + 0.977i)9-s + (−0.427 + 1.27i)10-s + (1.53 + 0.411i)11-s + (−0.276 + 0.961i)12-s + (0.560 − 0.150i)13-s + (−0.455 − 0.0284i)14-s + (0.543 − 1.22i)15-s + (−0.698 + 0.715i)16-s − 0.777·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.869 + 0.493i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.869 + 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953403 - 0.251536i\)
\(L(\frac12)\) \(\approx\) \(0.953403 - 0.251536i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.782i)T \)
3 \( 1 + (-1.34 - 1.08i)T \)
good5 \( 1 + (0.777 + 2.90i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.04 + 0.603i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.09 - 1.36i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.02 + 0.541i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 3.20T + 17T^{2} \)
19 \( 1 + (1.87 + 1.87i)T + 19iT^{2} \)
23 \( 1 + (3.61 + 2.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.12 - 7.94i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-1.39 + 2.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.10 - 5.10i)T - 37iT^{2} \)
41 \( 1 + (9.93 + 5.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.09 - 0.293i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.84 + 3.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.613 - 0.613i)T - 53iT^{2} \)
59 \( 1 + (-3.16 - 11.8i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.98 + 7.40i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-9.86 + 2.64i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.8iT - 71T^{2} \)
73 \( 1 + 2.87iT - 73T^{2} \)
79 \( 1 + (0.913 + 1.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.757 + 2.82i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 2.11iT - 89T^{2} \)
97 \( 1 + (3.06 + 5.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.84640357932734763833263333573, −11.89269570774107331278268763875, −10.87762280623795996052588351494, −9.709715689163249323970616226118, −8.702216888805624182083560285453, −8.481667414797546520677354125988, −6.96567617338326011684673258318, −4.63261961990864715600484760040, −3.76017866981751298945783564398, −1.65321553224379085030278840494, 1.93730393185029525247924250472, 3.70743364078062223427691606193, 6.26727608121737918762291161119, 6.79072532821348193472024779304, 7.970949362624683039370522802533, 8.782880064956271572907882081279, 9.870106205858241573671233183254, 11.19053607327623459477031084287, 11.81095293233446907271634444917, 13.65413642844837616114547720850

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