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Properties

Label 2-507-169.16-c1-0-0
Degree $2$
Conductor $507$
Sign $-0.951 - 0.306i$
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  + (1.45 + 0.117i)2-s + (−0.845 − 0.534i)3-s + (0.142 + 0.0230i)4-s + (−0.575 − 0.510i)5-s + (−1.17 − 0.879i)6-s + (−2.84 + 2.96i)7-s + (−2.63 − 0.650i)8-s + (0.428 + 0.903i)9-s + (−0.780 − 0.812i)10-s + (−0.187 + 0.395i)11-s + (−0.107 − 0.0954i)12-s + (−1.77 − 3.14i)13-s + (−4.50 + 3.99i)14-s + (0.214 + 0.739i)15-s + (−4.04 − 1.35i)16-s + (−3.36 + 3.50i)17-s + ⋯
L(s)  = 1  + (1.03 + 0.0833i)2-s + (−0.487 − 0.308i)3-s + (0.0710 + 0.0115i)4-s + (−0.257 − 0.228i)5-s + (−0.477 − 0.359i)6-s + (−1.07 + 1.12i)7-s + (−0.932 − 0.229i)8-s + (0.142 + 0.301i)9-s + (−0.246 − 0.256i)10-s + (−0.0566 + 0.119i)11-s + (−0.0310 − 0.0275i)12-s + (−0.491 − 0.871i)13-s + (−1.20 + 1.06i)14-s + (0.0552 + 0.190i)15-s + (−1.01 − 0.337i)16-s + (−0.816 + 0.850i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.951 - 0.306i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.951 - 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0315856 + 0.201138i\)
\(L(\frac12)\) \(\approx\) \(0.0315856 + 0.201138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.845 + 0.534i)T \)
13 \( 1 + (1.77 + 3.14i)T \)
good2 \( 1 + (-1.45 - 0.117i)T + (1.97 + 0.320i)T^{2} \)
5 \( 1 + (0.575 + 0.510i)T + (0.602 + 4.96i)T^{2} \)
7 \( 1 + (2.84 - 2.96i)T + (-0.281 - 6.99i)T^{2} \)
11 \( 1 + (0.187 - 0.395i)T + (-6.95 - 8.52i)T^{2} \)
17 \( 1 + (3.36 - 3.50i)T + (-0.684 - 16.9i)T^{2} \)
19 \( 1 + (0.991 - 1.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.72 - 2.98i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.31 + 0.590i)T + (28.6 + 4.65i)T^{2} \)
31 \( 1 + (-0.0172 + 0.142i)T + (-30.0 - 7.41i)T^{2} \)
37 \( 1 + (2.97 - 1.26i)T + (25.6 - 26.6i)T^{2} \)
41 \( 1 + (-4.86 - 3.07i)T + (17.5 + 37.0i)T^{2} \)
43 \( 1 + (-11.3 - 4.84i)T + (29.7 + 31.0i)T^{2} \)
47 \( 1 + (-4.60 + 12.1i)T + (-35.1 - 31.1i)T^{2} \)
53 \( 1 + (7.03 + 1.73i)T + (46.9 + 24.6i)T^{2} \)
59 \( 1 + (8.88 - 2.96i)T + (47.1 - 35.4i)T^{2} \)
61 \( 1 + (3.40 - 11.7i)T + (-51.5 - 32.6i)T^{2} \)
67 \( 1 + (-8.35 + 1.35i)T + (63.5 - 21.2i)T^{2} \)
71 \( 1 + (-0.390 + 9.69i)T + (-70.7 - 5.71i)T^{2} \)
73 \( 1 + (9.67 + 14.0i)T + (-25.8 + 68.2i)T^{2} \)
79 \( 1 + (0.551 - 1.45i)T + (-59.1 - 52.3i)T^{2} \)
83 \( 1 + (11.4 + 5.99i)T + (47.1 + 68.3i)T^{2} \)
89 \( 1 + (-3.05 - 5.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.879 + 4.30i)T + (-89.2 + 38.0i)T^{2} \)
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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

−11.72833022927261379585017189944, −10.54465728860933135164320723169, −9.472443533669731368621376830925, −8.732572302807042271381173435024, −7.49990088690604856101335423360, −6.17012341748700265716549761983, −5.85756920476539079383492417553, −4.79485579745253330639687170711, −3.65315769270026349077758163120, −2.48012330916592599225680525412, 0.086430324797621259289611174652, 2.82103351565996783716865919571, 3.94441037392419462582938326935, 4.50644869805647087884479912482, 5.68664635974052474050255580679, 6.73279928953204454179809931688, 7.31990936341774489443011421162, 9.133897793594858956755172508577, 9.528138251629502249711634472380, 10.90495897938225445683039824792

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