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Properties

Label 1-1259-1259.1016-r1-0-0
Degree $1$
Conductor $1259$
Sign $-0.368 + 0.929i$
Analytic cond. $135.298$
Root an. cond. $135.298$
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  + (0.618 − 0.785i)2-s + (−0.358 + 0.933i)3-s + (−0.235 − 0.971i)4-s + (0.917 + 0.398i)5-s + (0.511 + 0.859i)6-s + (−0.787 − 0.616i)7-s + (−0.909 − 0.416i)8-s + (−0.742 − 0.669i)9-s + (0.880 − 0.474i)10-s + (−0.790 + 0.612i)11-s + (0.991 + 0.129i)12-s + (−0.344 + 0.938i)13-s + (−0.971 + 0.237i)14-s + (−0.700 + 0.713i)15-s + (−0.889 + 0.456i)16-s + (0.995 − 0.0897i)17-s + ⋯
L(s)  = 1  + (0.618 − 0.785i)2-s + (−0.358 + 0.933i)3-s + (−0.235 − 0.971i)4-s + (0.917 + 0.398i)5-s + (0.511 + 0.859i)6-s + (−0.787 − 0.616i)7-s + (−0.909 − 0.416i)8-s + (−0.742 − 0.669i)9-s + (0.880 − 0.474i)10-s + (−0.790 + 0.612i)11-s + (0.991 + 0.129i)12-s + (−0.344 + 0.938i)13-s + (−0.971 + 0.237i)14-s + (−0.700 + 0.713i)15-s + (−0.889 + 0.456i)16-s + (0.995 − 0.0897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.368 + 0.929i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1016, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ -0.368 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4681050564 + 0.6890311499i\)
\(L(\frac12)\) \(\approx\) \(0.4681050564 + 0.6890311499i\)
\(L(1)\) \(\approx\) \(1.101402198 - 0.1445547639i\)
\(L(1)\) \(\approx\) \(1.101402198 - 0.1445547639i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (0.618 - 0.785i)T \)
3 \( 1 + (-0.358 + 0.933i)T \)
5 \( 1 + (0.917 + 0.398i)T \)
7 \( 1 + (-0.787 - 0.616i)T \)
11 \( 1 + (-0.790 + 0.612i)T \)
13 \( 1 + (-0.344 + 0.938i)T \)
17 \( 1 + (0.995 - 0.0897i)T \)
19 \( 1 + (0.999 + 0.0349i)T \)
23 \( 1 + (-0.283 - 0.959i)T \)
29 \( 1 + (0.721 - 0.691i)T \)
31 \( 1 + (0.693 - 0.720i)T \)
37 \( 1 + (-0.545 - 0.838i)T \)
41 \( 1 + (0.594 + 0.803i)T \)
43 \( 1 + (-0.626 - 0.779i)T \)
47 \( 1 + (0.999 - 0.0249i)T \)
53 \( 1 + (-0.902 + 0.430i)T \)
59 \( 1 + (-0.653 + 0.757i)T \)
61 \( 1 + (-0.520 + 0.854i)T \)
67 \( 1 + (-0.962 + 0.271i)T \)
71 \( 1 + (-0.979 - 0.203i)T \)
73 \( 1 + (0.996 - 0.0798i)T \)
79 \( 1 + (-0.634 - 0.773i)T \)
83 \( 1 + (-0.850 - 0.526i)T \)
89 \( 1 + (-0.557 + 0.829i)T \)
97 \( 1 + (-0.950 - 0.309i)T \)
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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

−20.84844500243794621015092465717, −19.87075405712547108465412477404, −18.848445888014195821314682619834, −18.11136054615657519417082721424, −17.588131365909654453512188340890, −16.77279344323739239717678501945, −16.057486801482881576855852125797, −15.413547344630013573733712455873, −13.99747098680043090639026425154, −13.847387573961478969069322397043, −12.775812419201210499823188155769, −12.5526973149912645542254997457, −11.70819487422890675402404982209, −10.39749970912910979858156018683, −9.48869157051212965862472218958, −8.45085748113896802768816095555, −7.84668939941736978973133775606, −6.89269441862819005269471640814, −6.024856681079513380471859337556, −5.47719273379826997373646333386, −5.06677930255057179680398076259, −3.07035549553692214295970855396, −2.887705500300613421077401836112, −1.38253220701309348939786572164, −0.144313542439929347616482194, 1.07422803843401860133846721121, 2.46628283364195151161127366576, 3.04446745404757786058012746823, 4.112006646778775443273846611513, 4.80262744538377846328124839650, 5.738298557278962783070964566610, 6.34369636962582196626703993968, 7.37683738454920409567942999480, 9.06496540208561772381357410063, 9.76861433926525860535944177464, 10.16555118644311994781992724204, 10.71687483280316841898981990301, 11.83526042695722606376143931036, 12.43160464799896892208983618930, 13.52975122774354626593823379037, 14.04700215146314085970022541601, 14.76422548852920975535845773964, 15.68350930873780668043260657388, 16.450125868213698779749749581405, 17.26101437336462697902600042382, 18.170451964102643181308624723708, 18.88067312244117935550951276315, 19.833407509717635960882966629222, 20.736488384534165564482677572861, 21.01187015363787070192964144639

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