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Properties

Label 2-12e2-144.13-c1-0-14
Degree $2$
Conductor $144$
Sign $0.834 - 0.550i$
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  + (1.06 + 0.927i)2-s + (1.71 − 0.222i)3-s + (0.280 + 1.98i)4-s + (−0.798 − 2.97i)5-s + (2.04 + 1.35i)6-s + (−1.78 + 1.02i)7-s + (−1.53 + 2.37i)8-s + (2.90 − 0.764i)9-s + (1.90 − 3.92i)10-s + (−0.446 − 0.119i)11-s + (0.923 + 3.33i)12-s + (−5.67 + 1.52i)13-s + (−2.85 − 0.553i)14-s + (−2.03 − 4.93i)15-s + (−3.84 + 1.11i)16-s + 0.0443·17-s + ⋯
L(s)  = 1  + (0.755 + 0.655i)2-s + (0.991 − 0.128i)3-s + (0.140 + 0.990i)4-s + (−0.357 − 1.33i)5-s + (0.833 + 0.553i)6-s + (−0.673 + 0.388i)7-s + (−0.543 + 0.839i)8-s + (0.966 − 0.254i)9-s + (0.603 − 1.24i)10-s + (−0.134 − 0.0360i)11-s + (0.266 + 0.963i)12-s + (−1.57 + 0.421i)13-s + (−0.763 − 0.147i)14-s + (−0.525 − 1.27i)15-s + (−0.960 + 0.278i)16-s + 0.0107·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.834 - 0.550i$
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: \((2,\ 144,\ (\ :1/2),\ 0.834 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75840 + 0.527855i\)
\(L(\frac12)\) \(\approx\) \(1.75840 + 0.527855i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 - 0.927i)T \)
3 \( 1 + (-1.71 + 0.222i)T \)
good5 \( 1 + (0.798 + 2.97i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.78 - 1.02i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.446 + 0.119i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (5.67 - 1.52i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 0.0443T + 17T^{2} \)
19 \( 1 + (1.10 + 1.10i)T + 19iT^{2} \)
23 \( 1 + (-7.89 - 4.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.86 + 6.95i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-0.542 + 0.939i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.769 - 0.769i)T - 37iT^{2} \)
41 \( 1 + (5.77 + 3.33i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-11.0 - 2.96i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.22 + 2.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.44 + 2.44i)T - 53iT^{2} \)
59 \( 1 + (0.962 + 3.59i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.318 + 1.18i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (5.52 - 1.48i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.88iT - 71T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 + (-3.46 - 6.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.157 + 0.588i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 5.30iT - 89T^{2} \)
97 \( 1 + (5.88 + 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.08856171313936671941434149241, −12.64539954820110098123998793120, −11.75894864392110118613010967616, −9.585723769189810612836594759013, −8.895717241272941116910993299465, −7.86273368262414973114206604382, −6.89497218950580282954275723589, −5.22044003590904952738525474309, −4.21910820727539376021317310789, −2.69001966247847737214130037482, 2.65176030441082794634860919120, 3.33046801566824706976041870422, 4.76668315542086933062466271903, 6.71000544395643262465349203492, 7.38910763547471824639843667766, 9.174353259424972691181491488435, 10.32165729104696960497043010294, 10.68008909330716283886952745779, 12.25079161818982275618428403595, 13.02021170700659216596490170556

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