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Properties

Label 2-1008-252.187-c1-0-35
Degree $2$
Conductor $1008$
Sign $0.952 + 0.303i$
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  + (1.72 − 0.164i)3-s + 2.22i·5-s + (0.517 − 2.59i)7-s + (2.94 − 0.567i)9-s − 3.59i·11-s + (0.699 − 0.404i)13-s + (0.365 + 3.82i)15-s + (1.37 − 0.796i)17-s + (1.63 − 2.83i)19-s + (0.466 − 4.55i)21-s + 1.68i·23-s + 0.0687·25-s + (4.98 − 1.46i)27-s + (−2.13 + 3.69i)29-s + (−0.630 + 1.09i)31-s + ⋯
L(s)  = 1  + (0.995 − 0.0949i)3-s + 0.993i·5-s + (0.195 − 0.980i)7-s + (0.981 − 0.189i)9-s − 1.08i·11-s + (0.194 − 0.112i)13-s + (0.0942 + 0.988i)15-s + (0.334 − 0.193i)17-s + (0.375 − 0.649i)19-s + (0.101 − 0.994i)21-s + 0.352i·23-s + 0.0137·25-s + (0.959 − 0.281i)27-s + (−0.395 + 0.685i)29-s + (−0.113 + 0.196i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.952 + 0.303i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.952 + 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.417581045\)
\(L(\frac12)\) \(\approx\) \(2.417581045\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.164i)T \)
7 \( 1 + (-0.517 + 2.59i)T \)
good5 \( 1 - 2.22iT - 5T^{2} \)
11 \( 1 + 3.59iT - 11T^{2} \)
13 \( 1 + (-0.699 + 0.404i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.37 + 0.796i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.63 + 2.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.68iT - 23T^{2} \)
29 \( 1 + (2.13 - 3.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.630 - 1.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.94 + 6.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.57 - 4.37i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.47 - 4.31i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.332 - 0.576i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.27 - 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.15 + 3.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.44 - 4.29i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0888 + 0.0512i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (5.30 - 3.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.73 + 2.15i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.33 + 9.23i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.69 - 0.981i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.3 + 8.84i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.904256012549876105934210074629, −9.081892079662509689336358472396, −8.185374326674807387188691703395, −7.36228829900392394612972700079, −6.88850156136603645630615585038, −5.73117236578961775901330158226, −4.34602105704763416221215674529, −3.39378866083512914108172815352, −2.77533501632351448420437160768, −1.17211172127208161048503406789, 1.52408127684959928879759650201, 2.45096777398597183959500021048, 3.77237033642248481966506861827, 4.70916871306641868541708165796, 5.49177415759404401131807697558, 6.73429837681107909221296141959, 7.85058381446526687093042505478, 8.353750752821709598894005942071, 9.203555470497777387481383410081, 9.660744901479750935838128016905

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