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Properties

Label 1-507-507.395-r0-0-0
Degree $1$
Conductor $507$
Sign $-0.284 - 0.958i$
Analytic cond. $2.35449$
Root an. cond. $2.35449$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.239 − 0.970i)2-s + (−0.885 + 0.464i)4-s + (−0.822 + 0.568i)5-s + (−0.663 − 0.748i)7-s + (0.663 + 0.748i)8-s + (0.748 + 0.663i)10-s + (−0.239 + 0.970i)11-s + (−0.568 + 0.822i)14-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s − i·19-s + (0.464 − 0.885i)20-s + 22-s + 23-s + (0.354 − 0.935i)25-s + ⋯
L(s)  = 1  + (−0.239 − 0.970i)2-s + (−0.885 + 0.464i)4-s + (−0.822 + 0.568i)5-s + (−0.663 − 0.748i)7-s + (0.663 + 0.748i)8-s + (0.748 + 0.663i)10-s + (−0.239 + 0.970i)11-s + (−0.568 + 0.822i)14-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s − i·19-s + (0.464 − 0.885i)20-s + 22-s + 23-s + (0.354 − 0.935i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.284 - 0.958i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ -0.284 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3803817724 - 0.5094752783i\)
\(L(\frac12)\) \(\approx\) \(0.3803817724 - 0.5094752783i\)
\(L(1)\) \(\approx\) \(0.5984011112 - 0.2851144913i\)
\(L(1)\) \(\approx\) \(0.5984011112 - 0.2851144913i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.239 - 0.970i)T \)
5 \( 1 + (-0.822 + 0.568i)T \)
7 \( 1 + (-0.663 - 0.748i)T \)
11 \( 1 + (-0.239 + 0.970i)T \)
17 \( 1 + (-0.748 + 0.663i)T \)
19 \( 1 - iT \)
23 \( 1 + T \)
29 \( 1 + (0.970 - 0.239i)T \)
31 \( 1 + (-0.935 + 0.354i)T \)
37 \( 1 + (0.935 - 0.354i)T \)
41 \( 1 + (-0.992 + 0.120i)T \)
43 \( 1 + (0.354 - 0.935i)T \)
47 \( 1 + (0.464 - 0.885i)T \)
53 \( 1 + (0.748 - 0.663i)T \)
59 \( 1 + (0.822 - 0.568i)T \)
61 \( 1 + (-0.748 - 0.663i)T \)
67 \( 1 + (0.464 - 0.885i)T \)
71 \( 1 + (-0.992 + 0.120i)T \)
73 \( 1 + (0.239 - 0.970i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (0.992 + 0.120i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.822 - 0.568i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.94840049560481701735886095062, −23.2016950683074817293063353857, −22.4244735340700256045267145194, −21.553331849496738960808753299058, −20.33973615745930052910494338117, −19.3047444149851447710713511063, −18.812194506687011910608007405030, −17.95716852316075826915225689348, −16.61615019355125285911604028253, −16.28967342242973136496249166865, −15.49162439574002425472121853142, −14.73528202411593102604054989537, −13.5266956535560236230753104446, −12.830894064146674091481664503006, −11.817315723717386650851639310489, −10.714170077131350160052977032235, −9.42483917264283839560641323338, −8.777009598305575915290199808642, −8.03582577043563679569564377385, −7.00102025916069891478091677000, −5.98423769538278385217552500592, −5.16002647566160804219341369956, −4.08620439998213406133917077598, −2.93669622968620015305141266148, −0.955327876374611217433816512, 0.49800928622311774823489668728, 2.1454956127537395215192790865, 3.18640430748606240426529558365, 4.05491596251787903493968184353, 4.885588878604106880046011447820, 6.73602744313863190254066439016, 7.37098840102444626394056960502, 8.49591393592965683956645276085, 9.49991449726875844611295326220, 10.499094311533330206696565881127, 10.96856379092623994940314838562, 12.03988438731284457622168398447, 12.87833173322096184085243579660, 13.59326430126543070157142215000, 14.78943261257524421827318750965, 15.60303333440332930239892368704, 16.759978603377376329574560844432, 17.62174163535110638070232414868, 18.415242402544291654856345412061, 19.47612483243422521300637958352, 19.79681956806793321912147638330, 20.57859667798848101657508943937, 21.78233988302862279861660065438, 22.41847757138298987872136993067, 23.32169863398042762696292586855

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