Properties

Label 1-340-340.99-r0-0-0
Degree $1$
Conductor $340$
Sign $-0.250 + 0.968i$
Analytic cond. $1.57895$
Root an. cond. $1.57895$
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  + (−0.923 − 0.382i)3-s + (0.382 + 0.923i)7-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s − i·13-s + (−0.707 + 0.707i)19-s − i·21-s + (−0.923 + 0.382i)23-s + (−0.382 − 0.923i)27-s + (−0.382 + 0.923i)29-s + (−0.923 − 0.382i)31-s + 33-s + (0.923 + 0.382i)37-s + (−0.382 + 0.923i)39-s + (0.382 + 0.923i)41-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (0.382 + 0.923i)7-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s − i·13-s + (−0.707 + 0.707i)19-s − i·21-s + (−0.923 + 0.382i)23-s + (−0.382 − 0.923i)27-s + (−0.382 + 0.923i)29-s + (−0.923 − 0.382i)31-s + 33-s + (0.923 + 0.382i)37-s + (−0.382 + 0.923i)39-s + (0.382 + 0.923i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(340\)    =    \(2^{2} \cdot 5 \cdot 17\)
Sign: $-0.250 + 0.968i$
Analytic conductor: \(1.57895\)
Root analytic conductor: \(1.57895\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{340} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 340,\ (0:\ ),\ -0.250 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3391111385 + 0.4379316256i\)
\(L(\frac12)\) \(\approx\) \(0.3391111385 + 0.4379316256i\)
\(L(1)\) \(\approx\) \(0.6724339011 + 0.1101923700i\)
\(L(1)\) \(\approx\) \(0.6724339011 + 0.1101923700i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 - iT \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.923 + 0.382i)T \)
29 \( 1 + (-0.382 + 0.923i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 - T \)
71 \( 1 + (0.923 + 0.382i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.382 - 0.923i)T \)
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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

−24.12151450873077429039683552251, −23.87347507097200717087921792666, −23.030695692948661549114219551764, −21.968901962578746188464909630438, −21.22486090134570782135482833116, −20.489649321905621273251509561679, −19.25713779756883421186344966782, −18.24879784628978099281836229713, −17.44188879939395758069699385332, −16.58290127603522323610050704868, −15.94632030974219695314620381041, −14.809538214666054332739194015397, −13.73708914583412371687013060927, −12.81076131117651496743780775383, −11.64016026935790140534908271746, −10.88361178420238512895877180040, −10.1948133451072055945937009031, −9.02417244574501411670263718089, −7.696406113535344658416062527159, −6.74526499768223306361927292403, −5.68496172143651992556324209681, −4.58572330848857002071620618411, −3.84380716841406603679537980484, −2.04281416026781247268099023705, −0.38459456590251412277973781877, 1.56535141577258544360346760095, 2.71384627681524871382830732488, 4.444895260612975128954279168146, 5.514043482890730586511766422982, 6.0517212163467364379601922105, 7.53886211954040695356726069708, 8.17556166620635024066138656762, 9.66277558238356580622643221523, 10.658226989677296032741268483267, 11.45420622334252154844124656431, 12.64342307441818255163287815052, 12.87468574847467244628367098595, 14.44621713101228732681592979782, 15.43044828346021889359350396922, 16.19740305824266617419105060579, 17.30579935870046167846935868884, 18.21079358816127266771903097437, 18.51024377824070977350784665764, 19.8194158532773693412864164607, 20.9277954900505548999580228906, 21.793955114945059727665035470207, 22.55752111939210023891467861704, 23.47323499136797105123184823175, 24.16472946880552854955813374270, 25.13144986314763451552113352906

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