Properties

Label 1-2675-2675.609-r1-0-0
Degree $1$
Conductor $2675$
Sign $-0.275 - 0.961i$
Analytic cond. $287.468$
Root an. cond. $287.468$
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  + (0.910 + 0.413i)2-s + (−0.563 − 0.826i)3-s + (0.657 + 0.753i)4-s + (−0.171 − 0.985i)6-s + (−0.430 + 0.902i)7-s + (0.286 + 0.958i)8-s + (−0.364 + 0.931i)9-s + (−0.275 + 0.961i)11-s + (0.252 − 0.967i)12-s + (−0.952 + 0.303i)13-s + (−0.765 + 0.643i)14-s + (−0.135 + 0.990i)16-s + (0.124 − 0.992i)17-s + (−0.717 + 0.696i)18-s + (−0.297 + 0.954i)19-s + ⋯
L(s)  = 1  + (0.910 + 0.413i)2-s + (−0.563 − 0.826i)3-s + (0.657 + 0.753i)4-s + (−0.171 − 0.985i)6-s + (−0.430 + 0.902i)7-s + (0.286 + 0.958i)8-s + (−0.364 + 0.931i)9-s + (−0.275 + 0.961i)11-s + (0.252 − 0.967i)12-s + (−0.952 + 0.303i)13-s + (−0.765 + 0.643i)14-s + (−0.135 + 0.990i)16-s + (0.124 − 0.992i)17-s + (−0.717 + 0.696i)18-s + (−0.297 + 0.954i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $-0.275 - 0.961i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ -0.275 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4840644229 + 0.6423625022i\)
\(L(\frac12)\) \(\approx\) \(-0.4840644229 + 0.6423625022i\)
\(L(1)\) \(\approx\) \(1.005767087 + 0.4951646004i\)
\(L(1)\) \(\approx\) \(1.005767087 + 0.4951646004i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (0.910 + 0.413i)T \)
3 \( 1 + (-0.563 - 0.826i)T \)
7 \( 1 + (-0.430 + 0.902i)T \)
11 \( 1 + (-0.275 + 0.961i)T \)
13 \( 1 + (-0.952 + 0.303i)T \)
17 \( 1 + (0.124 - 0.992i)T \)
19 \( 1 + (-0.297 + 0.954i)T \)
23 \( 1 + (-0.997 - 0.0710i)T \)
29 \( 1 + (0.461 + 0.886i)T \)
31 \( 1 + (-0.977 - 0.211i)T \)
37 \( 1 + (-0.194 + 0.980i)T \)
41 \( 1 + (-0.994 - 0.106i)T \)
43 \( 1 + (0.263 + 0.964i)T \)
47 \( 1 + (0.408 + 0.912i)T \)
53 \( 1 + (0.979 + 0.200i)T \)
59 \( 1 + (-0.986 - 0.165i)T \)
61 \( 1 + (0.725 - 0.687i)T \)
67 \( 1 + (0.999 - 0.0237i)T \)
71 \( 1 + (0.611 + 0.791i)T \)
73 \( 1 + (0.0769 + 0.997i)T \)
79 \( 1 + (-0.963 + 0.269i)T \)
83 \( 1 + (-0.966 + 0.257i)T \)
89 \( 1 + (-0.884 - 0.467i)T \)
97 \( 1 + (0.928 + 0.370i)T \)
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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

−18.859979551625023213835533768838, −17.77529459047003127887475441656, −16.958842947959181624515433641602, −16.49073901968688798885514346461, −15.65864610657079090786868577966, −15.15995250826982322284059256370, −14.29933155754943761806516596019, −13.67037081477534714848191467062, −12.8897682477425945271527339679, −12.181455789930161001029002971808, −11.434985940920695037666738455338, −10.614763108748061700354919958869, −10.347979448924412191803882188912, −9.59529673803080518034476932837, −8.57448222330656032288671440092, −7.3910145859589612613175700580, −6.62551771142549097308779271937, −5.82439302063063065085402749338, −5.27478505091124894745050285571, −4.30925295422463856713924027477, −3.795452399302230584778998680643, −3.07818662268176711558728850519, −2.05729961818944742520371414334, −0.649074593980010950234892824387, −0.1324931081187710342289706863, 1.61014924483990268953727715658, 2.32185349634185061299521692241, 2.9634599804390005188268147573, 4.22199671267784404276669114991, 5.10991749509715331569898605419, 5.52157382843214732760854713130, 6.46070680686952782088516169878, 6.98991555919489462468845116345, 7.71165399491043619080415262835, 8.42412565367648957538180172338, 9.571880014563286191208509427361, 10.37401626446120860445472069867, 11.54283073822818017992096463999, 11.95501333529497496403494619951, 12.64786227013205256582039364549, 12.8889005124579828671628913506, 14.13311691413418904654443561328, 14.4064210607721860304144480088, 15.4391174225494796131031857266, 16.03236900870562610342821339511, 16.76600268537271121226363099356, 17.369729868142884036051594683532, 18.3289917363915726516494662133, 18.628649312830853669243179223487, 19.82466536856748998389564153175

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