↑

Properties

Label 1-145-145.62-r1-0-0
Degree $1$
Conductor $145$
Sign $-0.408 + 0.912i$
Analytic cond. $15.5824$
Root an. cond. $15.5824$
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  + (−0.433 + 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (−0.900 + 0.433i)6-s + (0.781 + 0.623i)7-s + (0.974 − 0.222i)8-s + (0.222 + 0.974i)9-s + (0.222 − 0.974i)11-s − i·12-s + (0.974 + 0.222i)13-s + (−0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s − i·17-s + (−0.974 − 0.222i)18-s + (0.623 + 0.781i)19-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (−0.900 + 0.433i)6-s + (0.781 + 0.623i)7-s + (0.974 − 0.222i)8-s + (0.222 + 0.974i)9-s + (0.222 − 0.974i)11-s − i·12-s + (0.974 + 0.222i)13-s + (−0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s − i·17-s + (−0.974 − 0.222i)18-s + (0.623 + 0.781i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.408 + 0.912i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ -0.408 + 0.912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.120148813 + 1.728586417i\)
\(L(\frac12)\) \(\approx\) \(1.120148813 + 1.728586417i\)
\(L(1)\) \(\approx\) \(1.005780927 + 0.7762883138i\)
\(L(1)\) \(\approx\) \(1.005780927 + 0.7762883138i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.433 + 0.900i)T \)
3 \( 1 + (0.781 + 0.623i)T \)
7 \( 1 + (0.781 + 0.623i)T \)
11 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (0.974 + 0.222i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.623 + 0.781i)T \)
23 \( 1 + (0.433 + 0.900i)T \)
31 \( 1 + (0.900 + 0.433i)T \)
37 \( 1 + (-0.974 + 0.222i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (0.974 + 0.222i)T \)
53 \( 1 + (-0.433 + 0.900i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 + (0.974 - 0.222i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.433 - 0.900i)T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (0.781 - 0.623i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (0.781 - 0.623i)T \)
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

−27.82992324327344084960966920352, −26.58893020498352390307304651220, −26.01111506848214940304678155116, −24.90283581311253130874733841129, −23.705531150652310587580980392316, −22.735176522066576752364247670, −21.26128759483792116013526253096, −20.477826353136360297176066346602, −19.882790589872029946769861087817, −18.744643695657163459266232452884, −17.85653474495902027762116950638, −17.1167216236289592703178521765, −15.305095609169393354132796946256, −14.108930371793502281567541126818, −13.247211160795308659587652613455, −12.27896368595699662251094038561, −11.10415761357915510975264642398, −9.99052354227543582852185616884, −8.73876677140258806002053584409, −7.94959849589889195642474765781, −6.82695476512094405769180024623, −4.58509785816042288405679178480, −3.45125530866147843412545610287, −2.00582174285054746186021421961, −1.000685943028765683980576199753, 1.45407154241685756569551850616, 3.36563069823217138885594895395, 4.83755052167460442291023662884, 5.84657245306895074553131876851, 7.47520308161294618249675753370, 8.5398679966360614464043239328, 9.0926804852597299085228694022, 10.40193996241093848540924215388, 11.577092995224133716932177129488, 13.72363405423715872240504395867, 14.06392406874402731831713985837, 15.377444020480481244976501091139, 15.94154824110861795490254559175, 17.04688323838399788961505417167, 18.42967598145766962391855540202, 18.99725524952201024143262153978, 20.369330896450530623720714160090, 21.332474592036353529255114211945, 22.37570432233555653710543624171, 23.6473256306428084080292715865, 24.806254211169087770868200580091, 25.2097893617457403436224649559, 26.41988526767608839485196858971, 27.1944526080521625291061365065, 27.791024857021281612097456692184

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