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Properties

Label 2-2997-333.158-c0-0-1
Degree $2$
Conductor $2997$
Sign $0.748 + 0.663i$
Analytic cond. $1.49569$
Root an. cond. $1.22298$
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  + 4-s + (0.5 − 0.866i)7-s − 13-s + 16-s + (−1 − 1.73i)19-s + 25-s + (0.5 − 0.866i)28-s + (0.5 − 0.866i)31-s + 37-s + (0.5 + 0.866i)43-s − 52-s + (−1 + 1.73i)61-s + 64-s − 67-s − 73-s + ⋯
L(s)  = 1  + 4-s + (0.5 − 0.866i)7-s − 13-s + 16-s + (−1 − 1.73i)19-s + 25-s + (0.5 − 0.866i)28-s + (0.5 − 0.866i)31-s + 37-s + (0.5 + 0.866i)43-s − 52-s + (−1 + 1.73i)61-s + 64-s − 67-s − 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2997\)    =    \(3^{4} \cdot 37\)
Sign: $0.748 + 0.663i$
Analytic conductor: \(1.49569\)
Root analytic conductor: \(1.22298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2997} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2997,\ (\ :0),\ 0.748 + 0.663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.606894116\)
\(L(\frac12)\) \(\approx\) \(1.606894116\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 - T \)
good2 \( 1 - T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.804478726990770672691000312220, −7.80545026894614556749027200966, −7.34866689831626427902462001461, −6.69449001722593215744010835086, −5.94921081253932521749782155347, −4.77206375301230577038558256733, −4.31457968046287524402741190293, −2.92236174763292876432942603686, −2.34168194558921454342491896952, −1.02957966518131615499961513018, 1.59792292117289361494029362410, 2.35973887541789637363901757883, 3.18931086629007721691188813923, 4.37654365996118985262405650735, 5.29959636135619379803241804924, 5.99766395759242635947706755257, 6.69830142337303979952585525561, 7.55032693242450203079399462867, 8.182225453467164626832392892274, 8.857532697648804188234038417712

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