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Properties

Label 2-507-169.64-c1-0-5
Degree $2$
Conductor $507$
Sign $-0.164 - 0.986i$
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  + (−1.16 − 0.142i)2-s + (−0.748 + 0.663i)3-s + (−0.593 − 0.146i)4-s + (1.79 + 3.42i)5-s + (0.969 − 0.669i)6-s + (2.92 + 1.10i)7-s + (2.87 + 1.09i)8-s + (0.120 − 0.992i)9-s + (−1.61 − 4.25i)10-s + (−2.60 + 0.316i)11-s + (0.541 − 0.284i)12-s + (3.28 + 1.48i)13-s + (−3.26 − 1.71i)14-s + (−3.61 − 1.36i)15-s + (−2.12 − 1.11i)16-s + (−0.611 + 1.61i)17-s + ⋯
L(s)  = 1  + (−0.827 − 0.100i)2-s + (−0.432 + 0.382i)3-s + (−0.296 − 0.0731i)4-s + (0.802 + 1.52i)5-s + (0.395 − 0.273i)6-s + (1.10 + 0.418i)7-s + (1.01 + 0.385i)8-s + (0.0401 − 0.330i)9-s + (−0.510 − 1.34i)10-s + (−0.786 + 0.0955i)11-s + (0.156 − 0.0820i)12-s + (0.911 + 0.411i)13-s + (−0.871 − 0.457i)14-s + (−0.932 − 0.353i)15-s + (−0.531 − 0.279i)16-s + (−0.148 + 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.164 - 0.986i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.574006 + 0.677581i\)
\(L(\frac12)\) \(\approx\) \(0.574006 + 0.677581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.748 - 0.663i)T \)
13 \( 1 + (-3.28 - 1.48i)T \)
good2 \( 1 + (1.16 + 0.142i)T + (1.94 + 0.478i)T^{2} \)
5 \( 1 + (-1.79 - 3.42i)T + (-2.84 + 4.11i)T^{2} \)
7 \( 1 + (-2.92 - 1.10i)T + (5.23 + 4.64i)T^{2} \)
11 \( 1 + (2.60 - 0.316i)T + (10.6 - 2.63i)T^{2} \)
17 \( 1 + (0.611 - 1.61i)T + (-12.7 - 11.2i)T^{2} \)
19 \( 1 + 7.31iT - 19T^{2} \)
23 \( 1 - 8.64T + 23T^{2} \)
29 \( 1 + (0.653 - 5.38i)T + (-28.1 - 6.94i)T^{2} \)
31 \( 1 + (7.24 - 5.00i)T + (10.9 - 28.9i)T^{2} \)
37 \( 1 + (-1.28 + 0.885i)T + (13.1 - 34.5i)T^{2} \)
41 \( 1 + (-1.66 - 1.88i)T + (-4.94 + 40.7i)T^{2} \)
43 \( 1 + (-0.399 + 0.579i)T + (-15.2 - 40.2i)T^{2} \)
47 \( 1 + (-0.852 - 3.46i)T + (-41.6 + 21.8i)T^{2} \)
53 \( 1 + (2.08 - 5.48i)T + (-39.6 - 35.1i)T^{2} \)
59 \( 1 + (-2.32 - 4.43i)T + (-33.5 + 48.5i)T^{2} \)
61 \( 1 + (3.51 + 9.27i)T + (-45.6 + 40.4i)T^{2} \)
67 \( 1 + (1.27 + 5.16i)T + (-59.3 + 31.1i)T^{2} \)
71 \( 1 + (6.15 + 6.94i)T + (-8.55 + 70.4i)T^{2} \)
73 \( 1 + (11.4 - 1.38i)T + (70.8 - 17.4i)T^{2} \)
79 \( 1 + (-6.55 + 1.61i)T + (69.9 - 36.7i)T^{2} \)
83 \( 1 + (-3.46 + 3.91i)T + (-10.0 - 82.3i)T^{2} \)
89 \( 1 + 3.17iT - 89T^{2} \)
97 \( 1 + (2.76 - 5.26i)T + (-55.1 - 79.8i)T^{2} \)
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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

−10.80093614644289242275097206457, −10.64565713771056316791510662123, −9.266813157200498425108062939105, −8.864677449890470943622485181564, −7.54025746457059239716732192067, −6.72382534756150270901679886756, −5.48594493160897833615592110260, −4.72780513166774309439442078027, −3.00876447020857075620727914660, −1.66815278707202378628044954458, 0.839622280187823916769332510350, 1.70989520049591508227626532416, 4.18860340389224365989264497044, 5.14671936020893164197066019642, 5.77169507277019521622869106624, 7.44697500180518079474417039929, 8.148432865869209228898477302293, 8.733370078125567854315051167840, 9.664290932614317179009892454245, 10.53609333064109223938257188405

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