Properties

Label 2-507-507.59-c1-0-21
Degree $2$
Conductor $507$
Sign $0.192 - 0.981i$
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  + (−1.46 + 0.925i)3-s + (0.320 + 1.97i)4-s + (5.25 + 0.105i)7-s + (1.28 − 2.71i)9-s + (−2.29 − 2.59i)12-s + (2.59 + 2.5i)13-s + (−3.79 + 1.26i)16-s + (1.25 − 4.68i)19-s + (−7.79 + 4.71i)21-s + (4.96 + 0.602i)25-s + (0.626 + 5.15i)27-s + (1.47 + 10.4i)28-s + (−6.12 + 7.82i)31-s + (5.76 + 1.66i)36-s + (−10.2 + 4.14i)37-s + ⋯
L(s)  = 1  + (−0.845 + 0.534i)3-s + (0.160 + 0.987i)4-s + (1.98 + 0.0400i)7-s + (0.428 − 0.903i)9-s + (−0.663 − 0.748i)12-s + (0.720 + 0.693i)13-s + (−0.948 + 0.316i)16-s + (0.287 − 1.07i)19-s + (−1.70 + 1.02i)21-s + (0.992 + 0.120i)25-s + (0.120 + 0.992i)27-s + (0.279 + 1.96i)28-s + (−1.10 + 1.40i)31-s + (0.960 + 0.278i)36-s + (−1.69 + 0.681i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.192 - 0.981i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.192 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08291 + 0.890951i\)
\(L(\frac12)\) \(\approx\) \(1.08291 + 0.890951i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.46 - 0.925i)T \)
13 \( 1 + (-2.59 - 2.5i)T \)
good2 \( 1 + (-0.320 - 1.97i)T^{2} \)
5 \( 1 + (-4.96 - 0.602i)T^{2} \)
7 \( 1 + (-5.25 - 0.105i)T + (6.99 + 0.281i)T^{2} \)
11 \( 1 + (-8.52 - 6.95i)T^{2} \)
17 \( 1 + (-0.684 + 16.9i)T^{2} \)
19 \( 1 + (-1.25 + 4.68i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-28.6 + 4.65i)T^{2} \)
31 \( 1 + (6.12 - 7.82i)T + (-7.41 - 30.0i)T^{2} \)
37 \( 1 + (10.2 - 4.14i)T + (26.6 - 25.6i)T^{2} \)
41 \( 1 + (-37.0 - 17.5i)T^{2} \)
43 \( 1 + (-0.560 - 1.31i)T + (-29.7 + 31.0i)T^{2} \)
47 \( 1 + (-31.1 - 35.1i)T^{2} \)
53 \( 1 + (-46.9 + 24.6i)T^{2} \)
59 \( 1 + (-35.4 + 47.1i)T^{2} \)
61 \( 1 + (4.23 + 14.6i)T + (-51.5 + 32.6i)T^{2} \)
67 \( 1 + (0.690 + 0.958i)T + (-21.2 + 63.5i)T^{2} \)
71 \( 1 + (5.71 + 70.7i)T^{2} \)
73 \( 1 + (3.03 + 16.5i)T + (-68.2 + 25.8i)T^{2} \)
79 \( 1 + (-1.66 - 4.38i)T + (-59.1 + 52.3i)T^{2} \)
83 \( 1 + (-68.3 - 47.1i)T^{2} \)
89 \( 1 + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-8.28 + 5.47i)T + (38.0 - 89.2i)T^{2} \)
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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

−11.06373963984777249697639559887, −10.74609073199458605112559877464, −9.054997508788421608782783490452, −8.574115463559197366331480334317, −7.42617757020965619369794884641, −6.63583834598964531732409821787, −5.12332799321669429445595015664, −4.63087882810162580030856114405, −3.44404092898657544913681034318, −1.63518010249240750913927543322, 1.10440904786385412352338475884, 1.98530967030496015005390240494, 4.30144669137368616383459513389, 5.41651208192688735329480906421, 5.70612794668599166647536483118, 7.07110476571757818889881857904, 7.88112849985898480701992509883, 8.802089299697719829384923070040, 10.30705178828174924600079151652, 10.79122956963732606359773435831

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