Properties

Label 2-507-507.158-c1-0-17
Degree $2$
Conductor $507$
Sign $0.146 - 0.989i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (−1.59 + 0.678i)3-s + (1.06 + 1.69i)4-s + (3.10 + 1.55i)7-s + (2.07 − 2.16i)9-s + (−2.85 − 1.96i)12-s + (2.59 − 2.5i)13-s + (−1.71 + 3.61i)16-s + (1.30 + 4.88i)19-s + (−6.00 − 0.363i)21-s + (−4.67 + 1.77i)25-s + (−1.84 + 4.85i)27-s + (0.698 + 6.91i)28-s + (−1.68 − 3.74i)31-s + (5.87 + 1.20i)36-s + (6.10 + 4.39i)37-s + ⋯
L(s)  = 1  + (−0.919 + 0.391i)3-s + (0.534 + 0.845i)4-s + (1.17 + 0.586i)7-s + (0.692 − 0.721i)9-s + (−0.822 − 0.568i)12-s + (0.720 − 0.693i)13-s + (−0.428 + 0.903i)16-s + (0.300 + 1.11i)19-s + (−1.31 − 0.0792i)21-s + (−0.935 + 0.354i)25-s + (−0.354 + 0.935i)27-s + (0.131 + 1.30i)28-s + (−0.302 − 0.672i)31-s + (0.979 + 0.200i)36-s + (1.00 + 0.722i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.146 - 0.989i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.146 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03086 + 0.889551i\)
\(L(\frac12)\) \(\approx\) \(1.03086 + 0.889551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.59 - 0.678i)T \)
13 \( 1 + (-2.59 + 2.5i)T \)
good2 \( 1 + (-1.06 - 1.69i)T^{2} \)
5 \( 1 + (4.67 - 1.77i)T^{2} \)
7 \( 1 + (-3.10 - 1.55i)T + (4.20 + 5.59i)T^{2} \)
11 \( 1 + (10.9 + 0.442i)T^{2} \)
17 \( 1 + (13.5 - 10.2i)T^{2} \)
19 \( 1 + (-1.30 - 4.88i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (24.5 - 15.4i)T^{2} \)
31 \( 1 + (1.68 + 3.74i)T + (-20.5 + 23.2i)T^{2} \)
37 \( 1 + (-6.10 - 4.39i)T + (11.7 + 35.0i)T^{2} \)
41 \( 1 + (29.5 + 28.4i)T^{2} \)
43 \( 1 + (1.17 - 7.22i)T + (-40.7 - 13.6i)T^{2} \)
47 \( 1 + (-38.6 - 26.6i)T^{2} \)
53 \( 1 + (-6.38 - 52.6i)T^{2} \)
59 \( 1 + (45.7 + 37.3i)T^{2} \)
61 \( 1 + (2.74 + 13.4i)T + (-56.1 + 23.9i)T^{2} \)
67 \( 1 + (4.93 - 1.11i)T + (60.5 - 28.7i)T^{2} \)
71 \( 1 + (-68.1 + 19.7i)T^{2} \)
73 \( 1 + (4.08 + 2.46i)T + (33.9 + 64.6i)T^{2} \)
79 \( 1 + (-10.1 + 5.31i)T + (44.8 - 65.0i)T^{2} \)
83 \( 1 + (-19.8 - 80.5i)T^{2} \)
89 \( 1 + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.43 + 7.56i)T + (-15.5 - 95.7i)T^{2} \)
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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

−11.34240043160105691088947211735, −10.49969768928660987242419305778, −9.393424167825062829348851307670, −8.167128722288316006222307061369, −7.70002328883940145345666903716, −6.31166244341371361770384888970, −5.59732713548600793518482983312, −4.45067722145035726902527114684, −3.36416309698216511977182776010, −1.69543968639960698450913019286, 1.01380214141599594027752722331, 2.07807308265438530968957821975, 4.27252677452986649650011767455, 5.15668094553530499749023251687, 6.06703355046443666586835032080, 6.98536005724469451377764180189, 7.68518125496035918896553196125, 8.992465684180599260590854759354, 10.17831785819559791209938837165, 10.93169589273726207587306772027

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