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Properties

Label 1-145-145.103-r1-0-0
Degree $1$
Conductor $145$
Sign $-0.630 - 0.776i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.630 - 0.776i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ -0.630 - 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.390104651 - 2.920988530i\)
\(L(\frac12)\) \(\approx\) \(1.390104651 - 2.920988530i\)
\(L(1)\) \(\approx\) \(1.563504538 - 1.207390012i\)
\(L(1)\) \(\approx\) \(1.563504538 - 1.207390012i\)

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

−28.45793849221921929150130439432, −26.77020336045224644637192697236, −26.14449260554272707396919111720, −25.522367491553697013878342725247, −24.475938447373760578832711065089, −23.42716263351227462754617684925, −22.53825751992188112612570175911, −21.29535657774222645267805467036, −20.73312715266629882720186406218, −19.496809949700664435992776464640, −18.44878639742276980029799379167, −16.80922248338889549845213282417, −16.00022294351709568365042288216, −15.121939460219900850798561618633, −14.1270221462478257166545092277, −13.16663350895456927297512578367, −12.474855678787409019006040548031, −10.66499786717061627155224907470, −9.39167630516568135806546517670, −8.21671805633610972719891972865, −7.197335984769336204475584589234, −6.0294193808627128317223042115, −4.42482201713998204792157248748, −3.51167958618677170964193424229, −2.27834028693999448066600244983, 0.858760484549227729981093289150, 2.82169118505347853862116801097, 3.118423413045211067614933568275, 4.82655209326657088060190948586, 6.191739076339106929574817327230, 7.478082903246491166312042093048, 9.02428344703781495702468281839, 9.93574255780786871555317781644, 11.14654516075862890034818613033, 12.65138301542592404806713572537, 13.23007526262462079022553292073, 14.05639241893355216554413692993, 15.51931174032599236211511321139, 15.80813839149745746049604077055, 18.07521223728700887569157234608, 18.97645043823874803926459176758, 19.78489511055690594150574404517, 20.66859684179690139989642610108, 21.5362088787779912056973429306, 22.66463746615652521696105745747, 23.581376778992677467751721765314, 24.71233555087241619160427474709, 25.48328944198225294939716532670, 26.5734981985901831374501529668, 27.78142330307608470051768460620

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