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Properties

Label 1-684-684.155-r1-0-0
Degree $1$
Conductor $684$
Sign $0.865 + 0.500i$
Analytic cond. $73.5060$
Root an. cond. $73.5060$
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  + (−0.766 + 0.642i)5-s − 7-s + (−0.5 + 0.866i)11-s + (0.939 − 0.342i)13-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)23-s + (0.173 − 0.984i)25-s + (−0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + (0.766 − 0.642i)35-s − 37-s + (0.173 + 0.984i)41-s + (−0.173 − 0.984i)43-s + (−0.939 + 0.342i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)5-s − 7-s + (−0.5 + 0.866i)11-s + (0.939 − 0.342i)13-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)23-s + (0.173 − 0.984i)25-s + (−0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + (0.766 − 0.642i)35-s − 37-s + (0.173 + 0.984i)41-s + (−0.173 − 0.984i)43-s + (−0.939 + 0.342i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.865 + 0.500i$
Analytic conductor: \(73.5060\)
Root analytic conductor: \(73.5060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (1:\ ),\ 0.865 + 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9973692657 + 0.2675931822i\)
\(L(\frac12)\) \(\approx\) \(0.9973692657 + 0.2675931822i\)
\(L(1)\) \(\approx\) \(0.7732720663 + 0.07890866895i\)
\(L(1)\) \(\approx\) \(0.7732720663 + 0.07890866895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−22.5493274578279652312896042857, −21.52536059491660842605594270909, −20.86555599644224805539570173002, −19.83812333706606702339621858229, −19.23125682898628695239706579026, −18.635091639164085653862909896620, −17.387822078260531563690464557261, −16.483401042781642784449507954896, −15.91089147661799950179073424437, −15.32743804844570933407693096862, −14.0101417604941939005998592115, −13.08634452897841496591907297124, −12.65812300973726292985955489278, −11.45889740661899665753221877663, −10.849164782444655735022433804263, −9.65715716164166487315743915197, −8.75427907711755127929780682007, −8.121342462094706709713892635688, −6.99193268829710082084894625634, −6.01582556726205031761524361032, −5.14088875564284048729222683490, −3.69203185349188789993004483088, −3.45261201007629966538036019332, −1.729842266518146334547211540542, −0.46462033130327794182410011229, 0.53915984103600583513992871089, 2.293878439440254791675532350518, 3.223604791929237691429194083773, 4.03988607763445367554080522830, 5.22387465386081586879705363642, 6.43690393283200228386916135252, 7.0941959222344845310508506473, 7.96888451458854996142486944428, 9.06190367143557216468957054170, 10.01641173798775387956837867016, 10.79839937740259265877430860695, 11.65787454808306410932930217279, 12.66163682828393951786041497325, 13.27971881436736720591513879335, 14.42000635805934843328560660068, 15.294552251924907488226277834, 15.9006040841223833093171172705, 16.60857137983836696630586458005, 17.92928248649216765486551461827, 18.54703712567998701696132699702, 19.18976555848951626002592643554, 20.30308615684817758328959749575, 20.61122489658713523441589290971, 22.084618583449235157453016606558, 22.7131451983064043158798282772

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