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Properties

Label 1-4235-4235.1704-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.528 + 0.848i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.995 − 0.0950i)2-s + (−0.5 − 0.866i)3-s + (0.981 + 0.189i)4-s + (0.415 + 0.909i)6-s + (−0.959 − 0.281i)8-s + (−0.5 + 0.866i)9-s + (−0.327 − 0.945i)12-s + (0.654 − 0.755i)13-s + (0.928 + 0.371i)16-s + (0.888 + 0.458i)17-s + (0.580 − 0.814i)18-s + (−0.888 + 0.458i)19-s + (−0.928 − 0.371i)23-s + (0.235 + 0.971i)24-s + (−0.723 + 0.690i)26-s + 27-s + ⋯
L(s)  = 1  + (−0.995 − 0.0950i)2-s + (−0.5 − 0.866i)3-s + (0.981 + 0.189i)4-s + (0.415 + 0.909i)6-s + (−0.959 − 0.281i)8-s + (−0.5 + 0.866i)9-s + (−0.327 − 0.945i)12-s + (0.654 − 0.755i)13-s + (0.928 + 0.371i)16-s + (0.888 + 0.458i)17-s + (0.580 − 0.814i)18-s + (−0.888 + 0.458i)19-s + (−0.928 − 0.371i)23-s + (0.235 + 0.971i)24-s + (−0.723 + 0.690i)26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.528 + 0.848i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1704, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.528 + 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3524573272 + 0.1957271250i\)
\(L(\frac12)\) \(\approx\) \(0.3524573272 + 0.1957271250i\)
\(L(1)\) \(\approx\) \(0.5180770460 - 0.1292411409i\)
\(L(1)\) \(\approx\) \(0.5180770460 - 0.1292411409i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.995 - 0.0950i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (0.888 + 0.458i)T \)
19 \( 1 + (-0.888 + 0.458i)T \)
23 \( 1 + (-0.928 - 0.371i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (0.327 - 0.945i)T \)
37 \( 1 + (-0.981 + 0.189i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.580 + 0.814i)T \)
53 \( 1 + (-0.928 + 0.371i)T \)
59 \( 1 + (0.995 - 0.0950i)T \)
61 \( 1 + (0.580 + 0.814i)T \)
67 \( 1 + (-0.580 + 0.814i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (0.786 - 0.618i)T \)
79 \( 1 + (-0.235 + 0.971i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (-0.0475 - 0.998i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−18.15617639375939599247602123216, −17.51614816661384637611207558052, −16.89954708611119891654353574620, −16.24614254273129257039094059104, −15.88002212979333105577691794603, −15.11517331166324015029943214601, −14.43919468739212084680303489181, −13.70341664336836719995626883333, −12.37763776315880882633196783762, −11.98150290109861379992161364849, −11.09696373535987091081959948254, −10.74959391453212215375131375002, −9.922049043716239109588419146439, −9.38270210261163187898060455406, −8.70485197138693151697917557944, −8.085911407677346468186383757464, −6.9685200743547483669726997667, −6.53300182498752714953408796573, −5.62037365764133412759621840462, −5.0590312127236162740277553446, −3.881394739808435711613217242089, −3.359890021367976635798748636380, −2.22267986405139767636858431706, −1.35739927766269942858794224625, −0.210162597934500472375052319051, 0.84925026323261694492158242588, 1.63216522282679940974755467969, 2.33777910464252583555270298648, 3.25630125500657325858546880412, 4.19566123453369023233101428108, 5.62430850356542893208590710185, 5.93624289439414144448265809457, 6.670101710376854871158001195745, 7.51189380662982811931063779974, 8.21533507828744423464069253336, 8.40838874534076302049105479481, 9.66331589999680114686417262808, 10.280418844538014971650332658136, 10.918834680933729555298803000803, 11.56365991321041531822900317507, 12.30278875593660735624373617983, 12.79067160584868020368934499964, 13.55293489143770229742796324534, 14.53217106296388622660822171282, 15.20975849936183209055981891397, 16.080369067089660793318238576021, 16.67493938882603548133921382666, 17.27359773500232146404285584117, 17.79242874744779774560748127190, 18.58938569817384441297195497185

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