Properties

Label 1-1016-1016.507-r0-0-0
Degree $1$
Conductor $1016$
Sign $1$
Analytic cond. $4.71828$
Root an. cond. $4.71828$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 7-s + 9-s + 11-s − 13-s − 15-s + 17-s + 19-s − 21-s + 23-s + 25-s − 27-s + 29-s − 31-s − 33-s + 35-s − 37-s + 39-s + 41-s − 43-s + 45-s − 47-s + 49-s − 51-s + 53-s + 55-s + ⋯
L(s)  = 1  − 3-s + 5-s + 7-s + 9-s + 11-s − 13-s − 15-s + 17-s + 19-s − 21-s + 23-s + 25-s − 27-s + 29-s − 31-s − 33-s + 35-s − 37-s + 39-s + 41-s − 43-s + 45-s − 47-s + 49-s − 51-s + 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1016\)    =    \(2^{3} \cdot 127\)
Sign: $1$
Analytic conductor: \(4.71828\)
Root analytic conductor: \(4.71828\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1016} (507, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1016,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.688208407\)
\(L(\frac12)\) \(\approx\) \(1.688208407\)
\(L(1)\) \(\approx\) \(1.173544436\)
\(L(1)\) \(\approx\) \(1.173544436\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
127 \( 1 \)
good3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−21.4962182897574536328354580111, −21.23911600243020199140709527366, −20.156206508807639705230449631428, −19.13771455605172250945642932925, −18.136903564698083237808286180746, −17.7373131918548057664220088648, −16.86493527887887657601550977118, −16.614314664633948579393164130157, −15.22262518509573188355232737999, −14.43293945699911674533138752919, −13.83787040288952746991755559065, −12.66898400660828274757596271231, −12.004058365695495309565346538588, −11.30228958660319146815978171031, −10.36813848566123162003596065011, −9.704715900678302392868889351287, −8.85417405210986642480688940639, −7.50720110373379362648098391047, −6.88856070309343276546399082794, −5.81591213874615316812052526106, −5.18062669021066411606291335154, −4.51733326187690357048974013507, −3.106366904704409574486914549771, −1.715625821656284657629063641168, −1.12412568757777392511668734613, 1.12412568757777392511668734613, 1.715625821656284657629063641168, 3.106366904704409574486914549771, 4.51733326187690357048974013507, 5.18062669021066411606291335154, 5.81591213874615316812052526106, 6.88856070309343276546399082794, 7.50720110373379362648098391047, 8.85417405210986642480688940639, 9.704715900678302392868889351287, 10.36813848566123162003596065011, 11.30228958660319146815978171031, 12.004058365695495309565346538588, 12.66898400660828274757596271231, 13.83787040288952746991755559065, 14.43293945699911674533138752919, 15.22262518509573188355232737999, 16.614314664633948579393164130157, 16.86493527887887657601550977118, 17.7373131918548057664220088648, 18.136903564698083237808286180746, 19.13771455605172250945642932925, 20.156206508807639705230449631428, 21.23911600243020199140709527366, 21.4962182897574536328354580111

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