Properties

Label 2-1008-21.17-c1-0-13
Degree $2$
Conductor $1008$
Sign $-0.851 + 0.524i$
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  + (−1.22 − 2.12i)5-s + (0.5 − 2.59i)7-s + (1.22 + 0.707i)11-s − 5.19i·13-s + (−2.44 + 4.24i)17-s + (−1.5 + 0.866i)19-s + (−4.89 + 2.82i)23-s + (−0.499 + 0.866i)25-s − 2.82i·29-s + (−1.5 − 0.866i)31-s + (−6.12 + 2.12i)35-s + (0.5 + 0.866i)37-s − 7.34·41-s + 43-s + (−6.12 − 10.6i)47-s + ⋯
L(s)  = 1  + (−0.547 − 0.948i)5-s + (0.188 − 0.981i)7-s + (0.369 + 0.213i)11-s − 1.44i·13-s + (−0.594 + 1.02i)17-s + (−0.344 + 0.198i)19-s + (−1.02 + 0.589i)23-s + (−0.0999 + 0.173i)25-s − 0.525i·29-s + (−0.269 − 0.155i)31-s + (−1.03 + 0.358i)35-s + (0.0821 + 0.142i)37-s − 1.14·41-s + 0.152·43-s + (−0.893 − 1.54i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.851 + 0.524i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.851 + 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9149405855\)
\(L(\frac12)\) \(\approx\) \(0.9149405855\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.22 - 0.707i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
17 \( 1 + (2.44 - 4.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.89 - 2.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.34T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (6.12 + 10.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.44 - 1.41i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.44 + 4.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 1.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.5 + 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.07iT - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.34T + 83T^{2} \)
89 \( 1 + (-2.44 - 4.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−9.782208234374930927187254987524, −8.421013604632034561035253170049, −8.208112671744732059566052660910, −7.24343485790854955537990323303, −6.18265152210481856790290321320, −5.14849547405912801082018804465, −4.21669067850493467362631780918, −3.55260931380509967147128746600, −1.75572287177399794240998431593, −0.40867639096624215351846639502, 1.97408396542527041998296691842, 2.95448379391483643644199071445, 4.08485114690419334502937453661, 5.03176727449148042668527168400, 6.32976192454681345308502559923, 6.78995797602136760161903947639, 7.76346701051354021874408705268, 8.804927246108834388422799489856, 9.286939047622158258450498212870, 10.38197458184988010405036188691

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