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Properties

Label 1-2013-2013.1136-r0-0-0
Degree $1$
Conductor $2013$
Sign $-0.207 - 0.978i$
Analytic cond. $9.34833$
Root an. cond. $9.34833$
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  − i·2-s − 4-s + (−0.809 − 0.587i)5-s + (−0.587 − 0.809i)7-s + i·8-s + (−0.587 + 0.809i)10-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + 16-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.809 + 0.587i)20-s + (0.587 + 0.809i)23-s + (0.309 + 0.951i)25-s + (−0.951 − 0.309i)26-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−0.809 − 0.587i)5-s + (−0.587 − 0.809i)7-s + i·8-s + (−0.587 + 0.809i)10-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + 16-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.809 + 0.587i)20-s + (0.587 + 0.809i)23-s + (0.309 + 0.951i)25-s + (−0.951 − 0.309i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.207 - 0.978i$
Analytic conductor: \(9.34833\)
Root analytic conductor: \(9.34833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (1136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (0:\ ),\ -0.207 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7269434943 - 0.8970403281i\)
\(L(\frac12)\) \(\approx\) \(0.7269434943 - 0.8970403281i\)
\(L(1)\) \(\approx\) \(0.6834084597 - 0.5260754317i\)
\(L(1)\) \(\approx\) \(0.6834084597 - 0.5260754317i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 - iT \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (0.587 + 0.809i)T \)
29 \( 1 + (0.587 - 0.809i)T \)
31 \( 1 + (-0.587 + 0.809i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (0.951 + 0.309i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + (0.951 + 0.309i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.587 - 0.809i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−19.81263835328410335189621856402, −19.18382820841082007285911750505, −18.57205331018262234064659776365, −18.1539472067640879528289284244, −17.02709209899814642637555145649, −16.31657420804540949530864195160, −15.808687545412569650569172285661, −15.093969063115004979809689731187, −14.51574573418425700293810441138, −13.79432043242010845433233694091, −12.7533767947656919970142512916, −12.24264686059910165050820863088, −11.29895979592856587500049965953, −10.41806059338862446333377007041, −9.43038208122894889482538330870, −8.828284743929060757556706797586, −8.09521711101949145038835717473, −7.15379324245132641439451202590, −6.667249708200084733522185153841, −5.84739464903878846686379805534, −5.01138907494002005514878507442, −3.96639016144806668672130950215, −3.3983711897603645232886937825, −2.28912430111912051065901112095, −0.68100964197870329300779978615, 0.7911473905604928436492229707, 1.2135411325775255968961192134, 2.88024891807142410883446297154, 3.37795749497224445595376013436, 4.1386570655874818167115648006, 5.0188305383245833032427000316, 5.74024865353838677998122313620, 7.15204434859216070723547483243, 7.84311829517386940017137538629, 8.52560748794176392240218587169, 9.58946310451837956338073751310, 9.989935446818847531764268261636, 10.94513452255710396100888763156, 11.63402361432273923940963171731, 12.28354258496351084010957861427, 13.08449971987779879170230736466, 13.53299548359121259922309790920, 14.43076134104342727644779724480, 15.39851322464127323301612108385, 16.218004237078077944871053463312, 16.82895543049692159405731980103, 17.68136185106353969143621374832, 18.48223860614443619148896420251, 19.17694519664132158614965381347, 20.00158937108959050929621031151

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