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Properties

Label 1-145-145.78-r1-0-0
Degree $1$
Conductor $145$
Sign $0.412 + 0.911i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.781 − 0.623i)2-s + (−0.974 − 0.222i)3-s + (0.222 + 0.974i)4-s + (0.623 + 0.781i)6-s + (0.974 + 0.222i)7-s + (0.433 − 0.900i)8-s + (0.900 + 0.433i)9-s + (−0.900 + 0.433i)11-s − i·12-s + (−0.433 − 0.900i)13-s + (−0.623 − 0.781i)14-s + (−0.900 + 0.433i)16-s − i·17-s + (−0.433 − 0.900i)18-s + (0.222 + 0.974i)19-s + ⋯
L(s)  = 1  + (−0.781 − 0.623i)2-s + (−0.974 − 0.222i)3-s + (0.222 + 0.974i)4-s + (0.623 + 0.781i)6-s + (0.974 + 0.222i)7-s + (0.433 − 0.900i)8-s + (0.900 + 0.433i)9-s + (−0.900 + 0.433i)11-s − i·12-s + (−0.433 − 0.900i)13-s + (−0.623 − 0.781i)14-s + (−0.900 + 0.433i)16-s − i·17-s + (−0.433 − 0.900i)18-s + (0.222 + 0.974i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.412 + 0.911i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ 0.412 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4281494590 + 0.2762689307i\)
\(L(\frac12)\) \(\approx\) \(0.4281494590 + 0.2762689307i\)
\(L(1)\) \(\approx\) \(0.5300507261 - 0.07242107819i\)
\(L(1)\) \(\approx\) \(0.5300507261 - 0.07242107819i\)

Euler product

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

−27.93174678761176552466225113486, −26.59316137076727904205513622478, −26.40510861169789215935191472947, −24.58008482159564189293690301620, −23.95115467523380930932643532521, −23.32632197358440135192984137807, −21.83185716741160420432247897733, −20.96918410051727107034828146923, −19.57100150866407423293727057350, −18.398624286751430813414888631253, −17.68244527589223838036376685807, −16.82690907547020101332177339441, −15.92217862007564799777477185857, −14.92507320058578737187856973489, −13.730300904699883049603114683068, −12.04008376684888819563911548707, −10.93915900954631121385342307202, −10.31367954389200040487273800959, −8.89326297376922863134409489820, −7.708274939924343827673547364106, −6.608719756789187443770436117089, −5.41975464477822216181310377039, −4.486966032229196826163015250140, −1.866666313967216172061034418415, −0.32302850187444847660026412846, 1.19939641398074439901633741937, 2.55808379460385837900008020397, 4.50062931685374209145119067487, 5.67104821261312322066848812825, 7.43352594091217729287579140804, 8.00949690977666874029309224372, 9.766990471913699999214243072898, 10.57286564522259795576833321935, 11.65348840829977693781007715564, 12.32604644745628788211055364116, 13.51697609452546515314200757326, 15.32608780725043426006419363508, 16.34420974882235078419294025185, 17.53526014616086087631078614367, 18.026662978433684954142174969809, 18.85171156596620139858141578180, 20.332491324404770366271726173816, 21.065946995892710913670600187, 22.1556481899350557919502631909, 23.07582547026501366683254091141, 24.376672585604902658127239360573, 25.18242100097234650993016897336, 26.57836223526227503815828713986, 27.57031251336241988748914451088, 27.990428300596873696127569979286

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