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Properties

Label 1-507-507.215-r0-0-0
Degree $1$
Conductor $507$
Sign $0.872 + 0.489i$
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  + (−0.534 + 0.845i)2-s + (−0.428 − 0.903i)4-s + (0.935 + 0.354i)5-s + (0.600 − 0.799i)7-s + (0.992 + 0.120i)8-s + (−0.799 + 0.600i)10-s + (0.999 − 0.0402i)11-s + (0.354 + 0.935i)14-s + (−0.632 + 0.774i)16-s + (0.799 + 0.600i)17-s + (−0.866 − 0.5i)19-s + (−0.0804 − 0.996i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.748 + 0.663i)25-s + ⋯
L(s)  = 1  + (−0.534 + 0.845i)2-s + (−0.428 − 0.903i)4-s + (0.935 + 0.354i)5-s + (0.600 − 0.799i)7-s + (0.992 + 0.120i)8-s + (−0.799 + 0.600i)10-s + (0.999 − 0.0402i)11-s + (0.354 + 0.935i)14-s + (−0.632 + 0.774i)16-s + (0.799 + 0.600i)17-s + (−0.866 − 0.5i)19-s + (−0.0804 − 0.996i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.748 + 0.663i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.872 + 0.489i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ 0.872 + 0.489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.306208237 + 0.3413367652i\)
\(L(\frac12)\) \(\approx\) \(1.306208237 + 0.3413367652i\)
\(L(1)\) \(\approx\) \(1.020871510 + 0.2722073441i\)
\(L(1)\) \(\approx\) \(1.020871510 + 0.2722073441i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.534 + 0.845i)T \)
5 \( 1 + (0.935 + 0.354i)T \)
7 \( 1 + (0.600 - 0.799i)T \)
11 \( 1 + (0.999 - 0.0402i)T \)
17 \( 1 + (0.799 + 0.600i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.845 + 0.534i)T \)
31 \( 1 + (-0.663 - 0.748i)T \)
37 \( 1 + (0.316 - 0.948i)T \)
41 \( 1 + (0.721 - 0.692i)T \)
43 \( 1 + (-0.948 + 0.316i)T \)
47 \( 1 + (-0.822 + 0.568i)T \)
53 \( 1 + (-0.120 + 0.992i)T \)
59 \( 1 + (0.774 - 0.632i)T \)
61 \( 1 + (-0.919 - 0.391i)T \)
67 \( 1 + (0.903 + 0.428i)T \)
71 \( 1 + (-0.960 - 0.278i)T \)
73 \( 1 + (0.464 - 0.885i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (-0.239 + 0.970i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.160 + 0.987i)T \)
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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

−23.42425422485724241987502611449, −22.341784569972782517862360379272, −21.51647076396370405732899114066, −21.19108188418059484937947053380, −20.20055428829446829482718539426, −19.34226350093786899580226793334, −18.3876084139618851223548554108, −17.76559627846161943669428622756, −16.993575995056135176453963088862, −16.21778899076225220265222487735, −14.74143557557423048330449925786, −13.98625165763459345139322104583, −13.00274962288141156758675844780, −12.05274284074552060749697367046, −11.54283835638406109116657865221, −10.25581727752760100988315462065, −9.5705092737025953503610715462, −8.75793482201359539275699175527, −8.00162488726242050602979086748, −6.58452230689965476284436153188, −5.44412715930051116990690356486, −4.459483122854332145543689226688, −3.16700019469335518806132685532, −1.99879490472123713028050259391, −1.30758186968942750887282638510, 1.07610638548409375405436990910, 2.079847151813992940620240963016, 3.89686723019838634877748044376, 4.91108487792030558897842070973, 6.078130315200710890917612939851, 6.67876882578888630013895153221, 7.69037532314118781395395793673, 8.67482011336530835903414494508, 9.57815070127433619639850243445, 10.45808768259110554777131736209, 11.08498475840331915781890508535, 12.65223917562633462285442729090, 13.73573030971843513634895585725, 14.46206963210781673779118472264, 14.82964997809103420436911159653, 16.315012799137238897152572887977, 17.005097985421612335470878574624, 17.55287049760555321503032780297, 18.34666363218899852633830940989, 19.326157046908726710039998515575, 20.13396280270398438426042250626, 21.24006327979012262519202497567, 22.10226079421696599286938458769, 23.02721401176277170889128086000, 23.84739924522520765907601272370

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