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Properties

Label 1-507-507.206-r0-0-0
Degree $1$
Conductor $507$
Sign $-0.337 + 0.941i$
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  + (−0.979 + 0.200i)2-s + (0.919 − 0.391i)4-s + (0.239 + 0.970i)5-s + (−0.903 − 0.428i)7-s + (−0.822 + 0.568i)8-s + (−0.428 − 0.903i)10-s + (0.316 + 0.948i)11-s + (0.970 + 0.239i)14-s + (0.692 − 0.721i)16-s + (0.428 − 0.903i)17-s + (0.866 − 0.5i)19-s + (0.600 + 0.799i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.885 + 0.464i)25-s + ⋯
L(s)  = 1  + (−0.979 + 0.200i)2-s + (0.919 − 0.391i)4-s + (0.239 + 0.970i)5-s + (−0.903 − 0.428i)7-s + (−0.822 + 0.568i)8-s + (−0.428 − 0.903i)10-s + (0.316 + 0.948i)11-s + (0.970 + 0.239i)14-s + (0.692 − 0.721i)16-s + (0.428 − 0.903i)17-s + (0.866 − 0.5i)19-s + (0.600 + 0.799i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.885 + 0.464i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.337 + 0.941i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ -0.337 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3894130818 + 0.5530946999i\)
\(L(\frac12)\) \(\approx\) \(0.3894130818 + 0.5530946999i\)
\(L(1)\) \(\approx\) \(0.6163099091 + 0.2326698606i\)
\(L(1)\) \(\approx\) \(0.6163099091 + 0.2326698606i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.979 + 0.200i)T \)
5 \( 1 + (0.239 + 0.970i)T \)
7 \( 1 + (-0.903 - 0.428i)T \)
11 \( 1 + (0.316 + 0.948i)T \)
17 \( 1 + (0.428 - 0.903i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.200 + 0.979i)T \)
31 \( 1 + (-0.464 + 0.885i)T \)
37 \( 1 + (0.534 + 0.845i)T \)
41 \( 1 + (0.160 - 0.987i)T \)
43 \( 1 + (0.845 + 0.534i)T \)
47 \( 1 + (-0.992 + 0.120i)T \)
53 \( 1 + (-0.568 - 0.822i)T \)
59 \( 1 + (-0.721 + 0.692i)T \)
61 \( 1 + (-0.996 - 0.0804i)T \)
67 \( 1 + (0.391 - 0.919i)T \)
71 \( 1 + (0.774 + 0.632i)T \)
73 \( 1 + (-0.663 + 0.748i)T \)
79 \( 1 + (0.120 + 0.992i)T \)
83 \( 1 + (0.935 + 0.354i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.960 + 0.278i)T \)
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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

−23.573663114402337879987009673353, −22.239359427369767926565436652631, −21.51323325204380912837859750592, −20.68386073317185473187294757477, −19.84066602633358977014878002496, −19.14269093643233287397077210002, −18.39731219943559629344849481625, −17.33452051023151120965447111110, −16.410792963982166541848918814707, −16.23925865167779792365714409338, −15.05459039208955770721367468276, −13.72908011917995145308240710307, −12.67207883793397740720964472825, −12.13220885329691631068085608617, −11.07965810198919801868334037867, −9.9164109813073640660142377185, −9.35696599628968015337074499903, −8.46742977012866287299123253948, −7.736974690454703258373105915605, −6.17557811405589186291675757035, −5.84632920616868980162861580758, −4.053695222769468102399143902000, −2.99508740762223090183070398782, −1.754624096631746548885103192864, −0.53030472541792356720778771570, 1.33974525456634262850038334814, 2.66273102756434419940788160694, 3.47748296058626073799027162808, 5.22614579477677518094861831146, 6.42162238625661369073702793804, 7.08214085038431355096862867144, 7.68346589660141205106913014898, 9.32115864001540695973495897473, 9.69870954842388368734081230704, 10.55691444044680309391259781502, 11.483028482456218247937362619831, 12.434974556759277264968690924623, 13.79063969519509310600739544698, 14.5312573584591574136369826234, 15.58658840768000188418813505954, 16.1772786006415668791282971063, 17.26164613698251717337937889378, 17.99195807895203961799853263009, 18.61731511065368085171022252420, 19.71103371907938153900556836000, 20.0500359669829619597768925897, 21.251711393075923249156953732154, 22.34375050934214186770152715855, 22.98958690994668292333580640952, 23.924550427041458151226059107912

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