Properties

Label 2-12e2-16.5-c1-0-8
Degree $2$
Conductor $144$
Sign $-0.762 + 0.646i$
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  + (0.167 − 1.40i)2-s + (−1.94 − 0.470i)4-s + (−1.74 − 1.74i)5-s − 2.55i·7-s + (−0.985 + 2.65i)8-s + (−2.74 + 2.16i)10-s + (−0.473 − 0.473i)11-s + (2.88 − 2.88i)13-s + (−3.59 − 0.428i)14-s + (3.55 + 1.82i)16-s + 6.44·17-s + (−4.55 + 4.55i)19-s + (2.57 + 4.22i)20-s + (−0.744 + 0.585i)22-s + 2.82i·23-s + ⋯
L(s)  = 1  + (0.118 − 0.992i)2-s + (−0.971 − 0.235i)4-s + (−0.782 − 0.782i)5-s − 0.966i·7-s + (−0.348 + 0.937i)8-s + (−0.869 + 0.684i)10-s + (−0.142 − 0.142i)11-s + (0.800 − 0.800i)13-s + (−0.959 − 0.114i)14-s + (0.889 + 0.457i)16-s + 1.56·17-s + (−1.04 + 1.04i)19-s + (0.576 + 0.944i)20-s + (−0.158 + 0.124i)22-s + 0.589i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.762 + 0.646i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.762 + 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.314036 - 0.855844i\)
\(L(\frac12)\) \(\approx\) \(0.314036 - 0.855844i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.167 + 1.40i)T \)
3 \( 1 \)
good5 \( 1 + (1.74 + 1.74i)T + 5iT^{2} \)
7 \( 1 + 2.55iT - 7T^{2} \)
11 \( 1 + (0.473 + 0.473i)T + 11iT^{2} \)
13 \( 1 + (-2.88 + 2.88i)T - 13iT^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 + (4.55 - 4.55i)T - 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (-3.07 + 3.07i)T - 29iT^{2} \)
31 \( 1 - 6.55T + 31T^{2} \)
37 \( 1 + (2.72 + 2.72i)T + 37iT^{2} \)
41 \( 1 - 0.788iT - 41T^{2} \)
43 \( 1 + (0.389 + 0.389i)T + 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (-2.57 - 2.57i)T + 53iT^{2} \)
59 \( 1 + (4 + 4i)T + 59iT^{2} \)
61 \( 1 + (4.38 - 4.38i)T - 61iT^{2} \)
67 \( 1 + (2.11 - 2.11i)T - 67iT^{2} \)
71 \( 1 + 5.11iT - 71T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 + 6.31T + 79T^{2} \)
83 \( 1 + (0.641 - 0.641i)T - 83iT^{2} \)
89 \( 1 - 6.31iT - 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−12.55284488256728438552173845842, −11.84295881077251250845464750517, −10.64302732003041052297957468516, −9.986053911724900287392982530937, −8.449484805358268105640339424903, −7.86034894709645600930220385552, −5.77987398155214629625033749665, −4.37773732068345913092906258582, −3.44849502041304755193107455722, −0.997158448227561168532921654383, 3.20098876937620292405561914398, 4.65242191979274275044560137817, 6.08752734652116803683931669880, 6.99716846993753948382958260167, 8.180227567285300556422694213978, 8.996052310497827851762835439476, 10.37007432506907078510159472961, 11.66555315684616281673936379917, 12.51375864411288029597293766131, 13.75738485444369699437941625796

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