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Properties

Label 2-1008-63.20-c1-0-33
Degree $2$
Conductor $1008$
Sign $0.961 + 0.274i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
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  + (1.69 − 0.354i)3-s + (0.895 + 1.55i)5-s + (−0.0213 − 2.64i)7-s + (2.74 − 1.20i)9-s + (2.07 + 1.20i)11-s + (4.23 − 2.44i)13-s + (2.06 + 2.31i)15-s − 3.66·17-s + 3.01i·19-s + (−0.973 − 4.47i)21-s + (−3.26 + 1.88i)23-s + (0.897 − 1.55i)25-s + (4.23 − 3.00i)27-s + (−5.68 − 3.28i)29-s + (−4.02 + 2.32i)31-s + ⋯
L(s)  = 1  + (0.978 − 0.204i)3-s + (0.400 + 0.693i)5-s + (−0.00808 − 0.999i)7-s + (0.916 − 0.400i)9-s + (0.627 + 0.362i)11-s + (1.17 − 0.678i)13-s + (0.533 + 0.596i)15-s − 0.888·17-s + 0.692i·19-s + (−0.212 − 0.977i)21-s + (−0.680 + 0.392i)23-s + (0.179 − 0.310i)25-s + (0.815 − 0.579i)27-s + (−1.05 − 0.609i)29-s + (−0.722 + 0.417i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.961 + 0.274i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.961 + 0.274i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.580326775\)
\(L(\frac12)\) \(\approx\) \(2.580326775\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.69 + 0.354i)T \)
7 \( 1 + (0.0213 + 2.64i)T \)
good5 \( 1 + (-0.895 - 1.55i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.07 - 1.20i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.23 + 2.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.66T + 17T^{2} \)
19 \( 1 - 3.01iT - 19T^{2} \)
23 \( 1 + (3.26 - 1.88i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.68 + 3.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.02 - 2.32i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.36T + 37T^{2} \)
41 \( 1 + (-4.04 - 6.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.48 + 6.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.56 + 4.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-7.29 - 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.81 + 5.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.285 - 0.493i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.96iT - 71T^{2} \)
73 \( 1 - 12.3iT - 73T^{2} \)
79 \( 1 + (-1.51 + 2.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.00 - 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 + (4.77 + 2.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.898656631588346086441211137332, −9.151612614559048455832259096925, −8.172946746098194372150784881700, −7.47310728707771251181631655520, −6.66099377868989373568236634552, −5.90856758141704454047460373017, −4.17992219126893602525236569468, −3.70817180152186438261650870842, −2.48683471158569121015349079150, −1.31098253603550832222784961208, 1.52216983089604859383902974766, 2.48962973789593612959594406256, 3.75336878314177623387695295377, 4.58370038268781136918015927374, 5.73907399620827360458396637834, 6.53206516481518009028982930263, 7.71657996308987685658998970738, 8.745334172149774097980392778694, 9.064623588500683836344475828743, 9.488495637488289589947905208821

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