Properties

Label 2-503-503.2-c1-0-39
Degree $2$
Conductor $503$
Sign $-0.739 + 0.673i$
Analytic cond. $4.01647$
Root an. cond. $2.00411$
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  + (−0.102 − 0.126i)2-s + (0.592 − 0.513i)3-s + (0.404 − 1.93i)4-s + (−0.990 − 3.12i)5-s + (−0.125 − 0.0222i)6-s + (1.03 − 3.01i)7-s + (−0.575 + 0.296i)8-s + (−0.342 + 2.36i)9-s + (−0.293 + 0.446i)10-s + (1.46 − 0.472i)11-s + (−0.751 − 1.35i)12-s + (−3.42 + 1.15i)13-s + (−0.487 + 0.178i)14-s + (−2.19 − 1.34i)15-s + (−3.51 − 1.54i)16-s + (5.57 + 2.78i)17-s + ⋯
L(s)  = 1  + (−0.0726 − 0.0894i)2-s + (0.342 − 0.296i)3-s + (0.202 − 0.965i)4-s + (−0.442 − 1.39i)5-s + (−0.0513 − 0.00909i)6-s + (0.392 − 1.13i)7-s + (−0.203 + 0.104i)8-s + (−0.114 + 0.786i)9-s + (−0.0928 + 0.141i)10-s + (0.440 − 0.142i)11-s + (−0.216 − 0.390i)12-s + (−0.950 + 0.320i)13-s + (−0.130 + 0.0476i)14-s + (−0.565 − 0.347i)15-s + (−0.879 − 0.385i)16-s + (1.35 + 0.675i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(503\)
Sign: $-0.739 + 0.673i$
Analytic conductor: \(4.01647\)
Root analytic conductor: \(2.00411\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{503} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 503,\ (\ :1/2),\ -0.739 + 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515794 - 1.33257i\)
\(L(\frac12)\) \(\approx\) \(0.515794 - 1.33257i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad503 \( 1 + (20.6 - 8.83i)T \)
good2 \( 1 + (0.102 + 0.126i)T + (-0.410 + 1.95i)T^{2} \)
3 \( 1 + (-0.592 + 0.513i)T + (0.430 - 2.96i)T^{2} \)
5 \( 1 + (0.990 + 3.12i)T + (-4.08 + 2.87i)T^{2} \)
7 \( 1 + (-1.03 + 3.01i)T + (-5.51 - 4.31i)T^{2} \)
11 \( 1 + (-1.46 + 0.472i)T + (8.91 - 6.44i)T^{2} \)
13 \( 1 + (3.42 - 1.15i)T + (10.3 - 7.87i)T^{2} \)
17 \( 1 + (-5.57 - 2.78i)T + (10.2 + 13.5i)T^{2} \)
19 \( 1 + (-0.0566 - 0.127i)T + (-12.7 + 14.1i)T^{2} \)
23 \( 1 + (1.57 - 5.45i)T + (-19.4 - 12.2i)T^{2} \)
29 \( 1 + (-1.13 + 9.53i)T + (-28.1 - 6.83i)T^{2} \)
31 \( 1 + (-0.368 - 4.52i)T + (-30.5 + 5.02i)T^{2} \)
37 \( 1 + (-0.0478 - 0.0604i)T + (-8.49 + 36.0i)T^{2} \)
41 \( 1 + (-0.653 + 3.32i)T + (-37.9 - 15.5i)T^{2} \)
43 \( 1 + (-1.20 + 6.58i)T + (-40.1 - 15.2i)T^{2} \)
47 \( 1 + (-6.01 + 0.225i)T + (46.8 - 3.52i)T^{2} \)
53 \( 1 + (-5.36 + 12.0i)T + (-35.4 - 39.4i)T^{2} \)
59 \( 1 + (-3.66 + 8.81i)T + (-41.5 - 41.8i)T^{2} \)
61 \( 1 + (-2.64 - 4.37i)T + (-28.2 + 54.0i)T^{2} \)
67 \( 1 + (10.0 - 6.15i)T + (30.3 - 59.7i)T^{2} \)
71 \( 1 + (1.17 + 7.44i)T + (-67.5 + 21.8i)T^{2} \)
73 \( 1 + (4.75 + 4.55i)T + (3.19 + 72.9i)T^{2} \)
79 \( 1 + (-0.594 - 10.5i)T + (-78.4 + 8.88i)T^{2} \)
83 \( 1 + (3.66 + 7.01i)T + (-47.3 + 68.1i)T^{2} \)
89 \( 1 + (-2.06 - 7.87i)T + (-77.5 + 43.6i)T^{2} \)
97 \( 1 + (1.82 + 7.73i)T + (-86.7 + 43.3i)T^{2} \)
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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

−10.41816631994600234809140544737, −9.812911272184693043634825472225, −8.744601089028236425943461955303, −7.86071891466062464029770860774, −7.20122538263874861753805543756, −5.68044119198269617361499807108, −4.89639028059329260171125150059, −3.93344558757116560884882946300, −1.91383924126950071401391237996, −0.879687830561931790629592535387, 2.70750087663154816707158237956, 3.06379390295218441125753940319, 4.31653805183564902775062554634, 5.85335151062765402321031876713, 6.93081273096132143512988786299, 7.58706489744029694893345188180, 8.553650558090246143489335696499, 9.375816749404422584205244973396, 10.36671515515708530713610788205, 11.44764885181658924356132142194

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