Properties

Label 2-3312-3312.1931-c0-0-1
Degree $2$
Conductor $3312$
Sign $0.976 + 0.216i$
Analytic cond. $1.65290$
Root an. cond. $1.28565$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.173 − 0.984i)3-s + (−0.173 + 0.984i)4-s + (−0.642 + 0.766i)6-s + (0.866 − 0.500i)8-s + (−0.939 + 0.342i)9-s + 12-s + (0.424 + 1.58i)13-s + (−0.939 − 0.342i)16-s + (0.866 + 0.499i)18-s + (−0.866 − 0.5i)23-s + (−0.642 − 0.766i)24-s + (−0.866 + 0.5i)25-s + (0.939 − 1.34i)26-s + (0.5 + 0.866i)27-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.173 − 0.984i)3-s + (−0.173 + 0.984i)4-s + (−0.642 + 0.766i)6-s + (0.866 − 0.500i)8-s + (−0.939 + 0.342i)9-s + 12-s + (0.424 + 1.58i)13-s + (−0.939 − 0.342i)16-s + (0.866 + 0.499i)18-s + (−0.866 − 0.5i)23-s + (−0.642 − 0.766i)24-s + (−0.866 + 0.5i)25-s + (0.939 − 1.34i)26-s + (0.5 + 0.866i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $0.976 + 0.216i$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3312} (1931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ 0.976 + 0.216i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6699014257\)
\(L(\frac12)\) \(\approx\) \(0.6699014257\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 0.766i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (0.866 - 0.5i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
13 \( 1 + (-0.424 - 1.58i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
29 \( 1 + (0.0451 - 0.168i)T + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.342 - 0.592i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + 1.28iT - T^{2} \)
73 \( 1 - 1.53iT - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−8.782133337351247659799913077741, −8.109933798068787377513600753025, −7.46063065933997924389623910957, −6.64093927836203151833464302290, −6.09976800814098445815586169319, −4.78409873131255446629879376247, −3.97705395478604783705498313516, −2.89485776933964944602895882256, −2.00113288517309157491406391223, −1.22512577691904591397740350150, 0.56029304836549434446870244296, 2.25496473793740403850289740773, 3.49568848160537608051219232732, 4.32067373089683628313788704777, 5.31052435136476236241437824360, 5.77333446759340898508103992055, 6.44712985092744679080157945123, 7.57214334623243522315296196867, 8.214731627080430518628112811466, 8.692002345517383536053354983941

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