Properties

Label 2-12e2-144.83-c1-0-15
Degree $2$
Conductor $144$
Sign $-0.121 + 0.992i$
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  + (−0.845 − 1.13i)2-s + (1.38 − 1.03i)3-s + (−0.571 + 1.91i)4-s + (0.310 − 1.15i)5-s + (−2.34 − 0.692i)6-s + (0.356 − 0.616i)7-s + (2.65 − 0.972i)8-s + (0.840 − 2.87i)9-s + (−1.57 + 0.627i)10-s + (−0.611 − 2.28i)11-s + (1.20 + 3.24i)12-s + (−1.31 + 4.90i)13-s + (−1.00 + 0.117i)14-s + (−0.773 − 1.92i)15-s + (−3.34 − 2.18i)16-s − 0.863i·17-s + ⋯
L(s)  = 1  + (−0.597 − 0.801i)2-s + (0.800 − 0.599i)3-s + (−0.285 + 0.958i)4-s + (0.138 − 0.517i)5-s + (−0.959 − 0.282i)6-s + (0.134 − 0.233i)7-s + (0.939 − 0.343i)8-s + (0.280 − 0.959i)9-s + (−0.498 + 0.198i)10-s + (−0.184 − 0.687i)11-s + (0.346 + 0.938i)12-s + (−0.364 + 1.36i)13-s + (−0.267 + 0.0314i)14-s + (−0.199 − 0.497i)15-s + (−0.836 − 0.547i)16-s − 0.209i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.121 + 0.992i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.121 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692164 - 0.781823i\)
\(L(\frac12)\) \(\approx\) \(0.692164 - 0.781823i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.845 + 1.13i)T \)
3 \( 1 + (-1.38 + 1.03i)T \)
good5 \( 1 + (-0.310 + 1.15i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.356 + 0.616i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.611 + 2.28i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.31 - 4.90i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + 0.863iT - 17T^{2} \)
19 \( 1 + (0.539 - 0.539i)T - 19iT^{2} \)
23 \( 1 + (0.689 - 0.398i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.22 - 8.31i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-4.18 + 2.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.95 - 6.95i)T - 37iT^{2} \)
41 \( 1 + (-3.17 - 5.49i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-12.0 + 3.22i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.31 - 2.27i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.87 + 8.87i)T + 53iT^{2} \)
59 \( 1 + (12.7 + 3.40i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.548 + 0.146i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (6.89 + 1.84i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.03iT - 71T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 + (-0.841 - 0.486i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.1 - 2.99i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 4.35T + 89T^{2} \)
97 \( 1 + (2.89 - 5.01i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.70373128379114866682871954589, −11.93882124967794900126295526385, −10.80792562896234207912187461466, −9.459843280746069008414330428159, −8.837905501942259506157587888153, −7.84950096775912308927084915514, −6.71574483869452035820168491091, −4.53131597602328803125186322377, −3.04268442776686231024988967569, −1.46831794713778460737617459928, 2.54224182245139843873422921550, 4.50621331014015791969450993908, 5.78090664109345463952440124650, 7.29517231088521444194629790140, 8.107113054917617338537497277309, 9.164437727808293565014536057457, 10.21191328967492780799973564819, 10.69607227051131949720106186136, 12.56372831459728731724226154729, 13.82516551674514406122898372120

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