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Properties

Label 1-145-145.13-r1-0-0
Degree $1$
Conductor $145$
Sign $-0.847 - 0.531i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.847 - 0.531i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ -0.847 - 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03945837984 + 0.1371610983i\)
\(L(\frac12)\) \(\approx\) \(-0.03945837984 + 0.1371610983i\)
\(L(1)\) \(\approx\) \(0.4202975298 + 0.1703019367i\)
\(L(1)\) \(\approx\) \(0.4202975298 + 0.1703019367i\)

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

−27.52178191434572875673818477430, −26.72887275250330873034442734604, −25.6038337768221573463320730599, −24.577596145146641243590010032437, −23.282563493610416597414348197504, −22.274632011477885595811621275405, −21.60898840277424873077095139942, −20.25738718605651530773753897298, −19.14946602696820901769921707656, −18.56657024727355793809932967288, −17.30575498943626330604357241191, −16.57881509258341447714108169464, −15.84034433451057963822426674081, −13.72213523815177604923027369485, −12.73442070023753924705479965986, −11.68072554840605173484175110653, −10.96062782107301539693609497739, −9.732135869892219030847313878656, −8.82497541007677562119351626354, −7.01567589254439551845692044893, −6.51348468162990532269309473858, −4.57032599015758535759294481957, −3.15727108954874582086354645165, −1.36182183354333617729414209310, −0.09108066052333699368954130522, 1.457603621653181160188371101622, 3.80586479404048665938309375013, 5.52254254640424573312017185927, 6.248871998976652436643219098028, 7.28791502683427857636647337073, 8.83047100537285856919890301025, 9.909046738473727290843203196191, 10.67646784489761263399444845148, 11.99057919851131305983844233826, 13.12038972422965978841946388682, 14.88280032739855557690470970003, 15.60908977447577218929562545308, 16.712102585295181309039050149422, 17.27442908747618318768432192449, 18.409727556303080534413665941854, 19.29202081317942726153894504736, 20.42970349473651212615905073257, 21.94300562487532784983555100202, 22.88459243728984852005128832736, 23.52795979210485541619153554406, 24.89471703937426101761182796303, 25.54588460822985624492925124666, 26.74792752825431189621660026154, 27.702584855477330573931538446812, 28.28061543358533199641048758757

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