↑

Properties

Label 1-307-307.114-r0-0-0
Degree $1$
Conductor $307$
Sign $0.177 - 0.984i$
Analytic cond. $1.42570$
Root an. cond. $1.42570$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.932 − 0.361i)2-s + (−0.982 − 0.183i)3-s + (0.739 − 0.673i)4-s + (0.739 + 0.673i)5-s + (−0.982 + 0.183i)6-s + (−0.602 − 0.798i)7-s + (0.445 − 0.895i)8-s + (0.932 + 0.361i)9-s + (0.932 + 0.361i)10-s + (−0.982 + 0.183i)11-s + (−0.850 + 0.526i)12-s + (0.0922 − 0.995i)13-s + (−0.850 − 0.526i)14-s + (−0.602 − 0.798i)15-s + (0.0922 − 0.995i)16-s + 17-s + ⋯
L(s)  = 1  + (0.932 − 0.361i)2-s + (−0.982 − 0.183i)3-s + (0.739 − 0.673i)4-s + (0.739 + 0.673i)5-s + (−0.982 + 0.183i)6-s + (−0.602 − 0.798i)7-s + (0.445 − 0.895i)8-s + (0.932 + 0.361i)9-s + (0.932 + 0.361i)10-s + (−0.982 + 0.183i)11-s + (−0.850 + 0.526i)12-s + (0.0922 − 0.995i)13-s + (−0.850 − 0.526i)14-s + (−0.602 − 0.798i)15-s + (0.0922 − 0.995i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(307\)
Sign: $0.177 - 0.984i$
Analytic conductor: \(1.42570\)
Root analytic conductor: \(1.42570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{307} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 307,\ (0:\ ),\ 0.177 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.296409865 - 1.083540662i\)
\(L(\frac12)\) \(\approx\) \(1.296409865 - 1.083540662i\)
\(L(1)\) \(\approx\) \(1.319207217 - 0.5556625125i\)
\(L(1)\) \(\approx\) \(1.319207217 - 0.5556625125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad307 \( 1 \)
good2 \( 1 + (0.932 - 0.361i)T \)
3 \( 1 + (-0.982 - 0.183i)T \)
5 \( 1 + (0.739 + 0.673i)T \)
7 \( 1 + (-0.602 - 0.798i)T \)
11 \( 1 + (-0.982 + 0.183i)T \)
13 \( 1 + (0.0922 - 0.995i)T \)
17 \( 1 + T \)
19 \( 1 + (0.0922 - 0.995i)T \)
23 \( 1 + (0.932 + 0.361i)T \)
29 \( 1 + (0.932 - 0.361i)T \)
31 \( 1 + (-0.982 + 0.183i)T \)
37 \( 1 + (-0.602 - 0.798i)T \)
41 \( 1 + (-0.602 + 0.798i)T \)
43 \( 1 + (0.445 - 0.895i)T \)
47 \( 1 + (0.0922 - 0.995i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.273 + 0.961i)T \)
61 \( 1 + (0.445 - 0.895i)T \)
67 \( 1 + (-0.850 + 0.526i)T \)
71 \( 1 + (0.445 + 0.895i)T \)
73 \( 1 + (0.0922 - 0.995i)T \)
79 \( 1 + (0.445 + 0.895i)T \)
83 \( 1 + (-0.602 + 0.798i)T \)
89 \( 1 + (-0.850 + 0.526i)T \)
97 \( 1 + (0.0922 + 0.995i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.340271786833047773325639031608, −24.373614466624383136694591853, −23.65322163038862088131038069787, −22.83595028719919435180050715871, −21.923194268305588086048077546108, −21.18941939673263541321972712344, −20.779806595127388809599868567925, −19.01117347874639526657273918002, −18.11367586934484204347779760556, −16.86686028777570653919199997191, −16.39860109135410315165307779920, −15.68520763810127038748712760275, −14.47530141838496855319484562716, −13.350069937853045693825120079507, −12.51236373844569620483715301266, −12.02716395403565684677950896999, −10.73732792056539644644896017509, −9.70530036248938375922483882257, −8.47437028856678489908845183408, −7.022276039526385879014543006635, −5.9747604479035178075162892265, −5.45838371041966894897012935745, −4.56023093842027076357705025986, −3.118583387936664314064038113567, −1.68791134109920253289372873761, 0.971569607825221985186781920244, 2.53239072589527952789649945295, 3.564434438801789608852405125118, 5.10444900149598384705355121089, 5.66066438680385479389924578226, 6.82405110225496481779817807520, 7.40576491765447363072909996650, 9.80215614219723617958552583236, 10.47625717359223055670809058188, 10.98227017006462801015256519829, 12.32253654028587861694459387688, 13.17895770933491867580971162478, 13.65460561962950485401727578242, 15.0167633535990917133129856248, 15.86199156959168582810819304948, 16.90270278811653903186454013811, 17.87473462356211513320307031013, 18.76407579580809765278231267870, 19.781799427580255629146106904163, 20.955279250194775456758872432045, 21.638422043607861989863489656640, 22.5822009815616798278323744446, 23.1487022189312798263959669677, 23.72097333877497815290430694678, 25.00395290538509179072289765282

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