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Properties

Label 1-145-145.114-r1-0-0
Degree $1$
Conductor $145$
Sign $0.0384 + 0.999i$
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  + (0.974 + 0.222i)2-s + (0.433 + 0.900i)3-s + (0.900 + 0.433i)4-s + (0.222 + 0.974i)6-s + (0.900 − 0.433i)7-s + (0.781 + 0.623i)8-s + (−0.623 + 0.781i)9-s + (−0.781 + 0.623i)11-s + i·12-s + (0.623 + 0.781i)13-s + (0.974 − 0.222i)14-s + (0.623 + 0.781i)16-s − i·17-s + (−0.781 + 0.623i)18-s + (0.433 − 0.900i)19-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s + (0.433 + 0.900i)3-s + (0.900 + 0.433i)4-s + (0.222 + 0.974i)6-s + (0.900 − 0.433i)7-s + (0.781 + 0.623i)8-s + (−0.623 + 0.781i)9-s + (−0.781 + 0.623i)11-s + i·12-s + (0.623 + 0.781i)13-s + (0.974 − 0.222i)14-s + (0.623 + 0.781i)16-s − i·17-s + (−0.781 + 0.623i)18-s + (0.433 − 0.900i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0384 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0384 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.0384 + 0.999i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ 0.0384 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.959517460 + 2.847924016i\)
\(L(\frac12)\) \(\approx\) \(2.959517460 + 2.847924016i\)
\(L(1)\) \(\approx\) \(2.081635896 + 1.110475997i\)
\(L(1)\) \(\approx\) \(2.081635896 + 1.110475997i\)

Euler product

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

−28.12696758121381798637762910304, −26.60029732375223139792973541391, −25.32310431998609416246463905594, −24.67155692298288041212690307527, −23.78822918720641152317322223237, −23.07914986362545544760537990292, −21.69660237516325147540856227417, −20.80900789248911368596670490557, −20.025462141580408938559034292674, −18.758755736590954432145442854350, −18.067584429865805687154177556407, −16.46754544483916651678700481574, −15.09823495763159004250727945205, −14.44767690336687810403450118251, −13.304881768711033177451863397134, −12.59946607077866706675357079353, −11.47877481971151582851905445399, −10.48157688451784524855374934732, −8.48475973639336755970036285673, −7.707145375820155171848099944251, −6.18831132242738319613215245106, −5.35379504366813787837411224518, −3.633158165132995503203298578722, −2.44558838398147757926595000398, −1.221131630012835729703035376549, 2.04692817851099033926885141029, 3.415512933991791013822480055093, 4.62658443543256345252684910790, 5.27015273872206360708582173421, 7.087999033090180123533879622376, 8.08474395291875940129742526400, 9.52754013401708318909433835495, 10.948284774640621735825492126939, 11.56310984553911336160036832319, 13.306094964414904314199403121265, 14.0043359324802163214923890506, 15.019509483185566372978011221416, 15.823253442969351736349405098329, 16.769662501933524274860930428065, 18.03359475150376662413232576872, 19.795439877453380209587652729656, 20.66998600592285357468191813379, 21.19603757892462593313327896515, 22.22151705731068689299118868235, 23.31538397576848790486517478753, 24.09526554913860082638283049030, 25.38053377400285621702532244251, 26.0904116424982824018911802804, 27.08984075153583370418099603213, 28.24150073473308999671403900581

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