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Properties

Label 2-3312-3312.1885-c0-0-5
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 + (0.342 − 0.939i)3-s + (−0.173 − 0.984i)4-s + (0.5 + 0.866i)6-s + (0.866 + 0.500i)8-s + (−0.766 − 0.642i)9-s + (−0.984 − 0.173i)12-s + (−1.75 + 0.469i)13-s + (−0.939 + 0.342i)16-s + (0.984 − 0.173i)18-s + (−0.866 − 0.5i)23-s + (0.766 − 0.642i)24-s + (−0.866 + 0.5i)25-s + (0.766 − 1.64i)26-s + (−0.866 + 0.500i)27-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (0.342 − 0.939i)3-s + (−0.173 − 0.984i)4-s + (0.5 + 0.866i)6-s + (0.866 + 0.500i)8-s + (−0.766 − 0.642i)9-s + (−0.984 − 0.173i)12-s + (−1.75 + 0.469i)13-s + (−0.939 + 0.342i)16-s + (0.984 − 0.173i)18-s + (−0.866 − 0.5i)23-s + (0.766 − 0.642i)24-s + (−0.866 + 0.5i)25-s + (0.766 − 1.64i)26-s + (−0.866 + 0.500i)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: $\chi_{3312} (1885, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ -0.976 + 0.216i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1635943125\)
\(L(\frac12)\) \(\approx\) \(0.1635943125\)
\(L(1)\) 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 \( 1 + (-0.342 + 0.939i)T \)
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 \( 1 + (1.75 - 0.469i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
29 \( 1 + (-0.296 + 1.10i)T + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + (0.984 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (1.32 + 0.766i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - 1.96iT - T^{2} \)
73 \( 1 + 0.347iT - T^{2} \)
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

−8.395187058141293521338498377271, −7.63305312368603706988197602647, −7.06350890952233055370022734259, −6.57505775479020718041166046503, −5.58807119774525372146555762449, −4.94898445503523884478552571509, −3.75293120468215017959856735317, −2.36957041680199823205411935628, −1.73959208903143435126266184317, −0.10498017270335067907947461051, 1.91821118432202842040528222464, 2.72251763498796513907982646509, 3.52254153976081857215669471801, 4.40315619898219117831399723644, 5.04649568391114723785631686850, 6.07394481736489061201854949117, 7.42519907154317166867212499666, 7.78869547128815884585514405922, 8.560148961227327704888914207458, 9.585502816993674329096084323352

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