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Properties

Label 2-1008-9.7-c1-0-25
Degree $2$
Conductor $1008$
Sign $0.399 + 0.916i$
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  + (−1.18 + 1.26i)3-s + (0.686 + 1.18i)5-s + (0.5 − 0.866i)7-s + (−0.186 − 2.99i)9-s + (2.18 − 3.78i)11-s + (−1 − 1.73i)13-s + (−2.31 − 0.543i)15-s − 4.37·17-s − 5·19-s + (0.5 + 1.65i)21-s + (−3.68 − 6.38i)23-s + (1.55 − 2.69i)25-s + (4.00 + 3.31i)27-s + (−1.37 + 2.37i)29-s + (1 + 1.73i)31-s + ⋯
L(s)  = 1  + (−0.684 + 0.728i)3-s + (0.306 + 0.531i)5-s + (0.188 − 0.327i)7-s + (−0.0620 − 0.998i)9-s + (0.659 − 1.14i)11-s + (−0.277 − 0.480i)13-s + (−0.597 − 0.140i)15-s − 1.06·17-s − 1.14·19-s + (0.109 + 0.361i)21-s + (−0.768 − 1.33i)23-s + (0.311 − 0.539i)25-s + (0.769 + 0.638i)27-s + (−0.254 + 0.441i)29-s + (0.179 + 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.399 + 0.916i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.399 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9074835849\)
\(L(\frac12)\) \(\approx\) \(0.9074835849\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 - 1.26i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.686 - 1.18i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.18 + 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.37T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (3.68 + 6.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.37 - 2.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (5.18 + 8.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.55 + 7.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.74T + 53T^{2} \)
59 \( 1 + (-3.55 - 6.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.05 + 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.55 - 13.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + 5.11T + 73T^{2} \)
79 \( 1 + (-6.05 + 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.74 - 4.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.25T + 89T^{2} \)
97 \( 1 + (4.55 - 7.89i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.18003358126041927300478196569, −8.886907730399941467997410156539, −8.486077552541742581199357587449, −6.92893658300723183552476015433, −6.39057440179189247757570021477, −5.57064046654025726029633112489, −4.44484388581840663847271241772, −3.70517479695938801032386557591, −2.39720366973919982559689031396, −0.44980496150069241603862583430, 1.52446479141954707100979751003, 2.26307694191964373997342136842, 4.23128876236959883134571761448, 4.90028721063857012513000524765, 5.96091345001901944815310780112, 6.66660838196899357486294182498, 7.48784548549818682095629848357, 8.424687026914905621088491607195, 9.339354453472194629875615960025, 10.00935243958602024511614603005

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