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Properties

Label 2-1008-28.3-c1-0-1
Degree $2$
Conductor $1008$
Sign $-0.311 - 0.950i$
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  + (−2.5 − 0.866i)7-s + 5.19i·13-s + (−0.5 − 0.866i)19-s + (−2.5 + 4.33i)25-s + (−5.5 + 9.52i)31-s + (5.5 + 9.52i)37-s − 1.73i·43-s + (5.5 + 4.33i)49-s + (−6 + 3.46i)61-s + (−13.5 − 7.79i)67-s + (1.5 + 0.866i)73-s + (−4.5 + 2.59i)79-s + (4.5 − 12.9i)91-s + 13.8i·97-s + (6.5 + 11.2i)103-s + ⋯
L(s)  = 1  + (−0.944 − 0.327i)7-s + 1.44i·13-s + (−0.114 − 0.198i)19-s + (−0.5 + 0.866i)25-s + (−0.987 + 1.71i)31-s + (0.904 + 1.56i)37-s − 0.264i·43-s + (0.785 + 0.618i)49-s + (−0.768 + 0.443i)61-s + (−1.64 − 0.952i)67-s + (0.175 + 0.101i)73-s + (−0.506 + 0.292i)79-s + (0.471 − 1.36i)91-s + 1.40i·97-s + (0.640 + 1.10i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.311 - 0.950i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8568356593\)
\(L(\frac12)\) \(\approx\) \(0.8568356593\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (5.5 - 9.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.5 + 7.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 - 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−10.13614736388699184611999122108, −9.337916803516921899084753846003, −8.793676199073621590426559522344, −7.54064108802384523592163516580, −6.82004046332287558642972975745, −6.12898311810324774963097120620, −4.92651568058343011153403778636, −3.94664402982279755801921646358, −2.99855073557405389434607987799, −1.56560138545122196908044741465, 0.38458328176277377067320820992, 2.30949448237584298808730273506, 3.29581960159746243522278691003, 4.30296672381606252601703277882, 5.72077218042240509216660539367, 6.02201112600387668781992759031, 7.30486701652472527214542029487, 7.984819082816364742023555531218, 8.984326977143893156218675450510, 9.748234781246634507104937762896

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