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Properties

Label 2-507-13.10-c1-0-8
Degree $2$
Conductor $507$
Sign $0.702 - 0.711i$
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  + (−2.21 + 1.28i)2-s + (−0.5 − 0.866i)3-s + (2.28 − 3.95i)4-s − 0.561i·5-s + (2.21 + 1.28i)6-s + (3.08 + 1.78i)7-s + 6.56i·8-s + (−0.499 + 0.866i)9-s + (0.719 + 1.24i)10-s + (−1.73 + i)11-s − 4.56·12-s − 9.12·14-s + (−0.486 + 0.280i)15-s + (−3.84 − 6.65i)16-s + (1.28 − 2.21i)17-s − 2.56i·18-s + ⋯
L(s)  = 1  + (−1.56 + 0.905i)2-s + (−0.288 − 0.499i)3-s + (1.14 − 1.97i)4-s − 0.251i·5-s + (0.905 + 0.522i)6-s + (1.16 + 0.673i)7-s + 2.31i·8-s + (−0.166 + 0.288i)9-s + (0.227 + 0.393i)10-s + (−0.522 + 0.301i)11-s − 1.31·12-s − 2.43·14-s + (−0.125 + 0.0724i)15-s + (−0.960 − 1.66i)16-s + (0.310 − 0.538i)17-s − 0.603i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.702 - 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.612303 + 0.255927i\)
\(L(\frac12)\) \(\approx\) \(0.612303 + 0.255927i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (2.21 - 1.28i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 0.561iT - 5T^{2} \)
7 \( 1 + (-3.08 - 1.78i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.73 - i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.28 + 2.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.972 + 0.561i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.84 - 4.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.56iT - 31T^{2} \)
37 \( 1 + (-2.97 + 1.71i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.21 - 1.28i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.219 + 0.379i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (-9.63 - 5.56i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.06 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.379 - 0.219i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.1 - 7i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.87iT - 73T^{2} \)
79 \( 1 - 9.56T + 79T^{2} \)
83 \( 1 - 9.12iT - 83T^{2} \)
89 \( 1 + (-11.3 + 6.56i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.84 + 2.21i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.81774852979569924506364770592, −9.988820380433400401719544755728, −8.815254266720033840451595568797, −8.447419428593932565464517554435, −7.48581874907980893937014822043, −6.85273883288309229155922255746, −5.59192438597954214102621499995, −5.00183407881278575081353542156, −2.31582799341852914681894039369, −1.08197688517649470736827724740, 0.903118071279828179056018451984, 2.36259700712080783233910407137, 3.67878274951395642826261104006, 4.90590418366153579172877211429, 6.49422558333374625063223148021, 7.73446155305829222216194821087, 8.194637586558895335113505887886, 9.156400396442700653367677245402, 10.19492366505295259586542301142, 10.65829342016888750166436349439

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