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Properties

Label 2-1008-63.5-c1-0-34
Degree $2$
Conductor $1008$
Sign $-0.907 + 0.419i$
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  + (−0.690 − 1.58i)3-s + (1.10 + 1.91i)5-s + (−0.234 − 2.63i)7-s + (−2.04 + 2.19i)9-s + (−0.157 − 0.0909i)11-s + (−2.50 − 1.44i)13-s + (2.27 − 3.07i)15-s + (−1.98 − 3.43i)17-s + (−0.867 − 0.500i)19-s + (−4.02 + 2.19i)21-s + (−4.86 + 2.80i)23-s + (0.0605 − 0.104i)25-s + (4.89 + 1.73i)27-s + (0.703 − 0.406i)29-s − 7.96i·31-s + ⋯
L(s)  = 1  + (−0.398 − 0.917i)3-s + (0.493 + 0.855i)5-s + (−0.0885 − 0.996i)7-s + (−0.682 + 0.731i)9-s + (−0.0475 − 0.0274i)11-s + (−0.694 − 0.400i)13-s + (0.587 − 0.793i)15-s + (−0.481 − 0.833i)17-s + (−0.198 − 0.114i)19-s + (−0.878 + 0.478i)21-s + (−1.01 + 0.585i)23-s + (0.0121 − 0.0209i)25-s + (0.942 + 0.334i)27-s + (0.130 − 0.0754i)29-s − 1.43i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.907 + 0.419i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.907 + 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7515360144\)
\(L(\frac12)\) \(\approx\) \(0.7515360144\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.690 + 1.58i)T \)
7 \( 1 + (0.234 + 2.63i)T \)
good5 \( 1 + (-1.10 - 1.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.157 + 0.0909i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.50 + 1.44i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.98 + 3.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.867 + 0.500i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.86 - 2.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.703 + 0.406i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.96iT - 31T^{2} \)
37 \( 1 + (-1.25 + 2.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.612 + 1.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.47 + 9.48i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.15T + 47T^{2} \)
53 \( 1 + (-1.75 + 1.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.55T + 59T^{2} \)
61 \( 1 - 8.05iT - 61T^{2} \)
67 \( 1 + 6.89T + 67T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (10.1 - 5.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + (7.19 + 12.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.11 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.01 + 1.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.01185346621559682238604200897, −8.648195238677713335496821875582, −7.52605172670404516620852004869, −7.18355410031509136859801312441, −6.32350058537086194025045602176, −5.51044469134924867399232780388, −4.31412454994468694710029245738, −2.93209744119878357305578935897, −1.99294349064840387166140040756, −0.33815969062903600649621035981, 1.79054142052381406789763239948, 3.10524158446649867936893336347, 4.46108582236810162965280537961, 5.02059297779837756538049651140, 5.92341771016398322086580782852, 6.60352902525782778663150478120, 8.247419041994317516707074118795, 8.768122553287058729114447157188, 9.553410929387352286154297469338, 10.11499244883557907158028591793

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