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Properties

Label 1-644-644.107-r0-0-0
Degree $1$
Conductor $644$
Sign $0.945 - 0.325i$
Analytic cond. $2.99072$
Root an. cond. $2.99072$
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  + (0.327 + 0.945i)3-s + (−0.928 + 0.371i)5-s + (−0.786 + 0.618i)9-s + (0.235 − 0.971i)11-s + (0.415 − 0.909i)13-s + (−0.654 − 0.755i)15-s + (−0.0475 − 0.998i)17-s + (0.0475 − 0.998i)19-s + (0.723 − 0.690i)25-s + (−0.841 − 0.540i)27-s + (0.841 − 0.540i)29-s + (−0.981 + 0.189i)31-s + (0.995 − 0.0950i)33-s + (0.786 − 0.618i)37-s + (0.995 + 0.0950i)39-s + ⋯
L(s)  = 1  + (0.327 + 0.945i)3-s + (−0.928 + 0.371i)5-s + (−0.786 + 0.618i)9-s + (0.235 − 0.971i)11-s + (0.415 − 0.909i)13-s + (−0.654 − 0.755i)15-s + (−0.0475 − 0.998i)17-s + (0.0475 − 0.998i)19-s + (0.723 − 0.690i)25-s + (−0.841 − 0.540i)27-s + (0.841 − 0.540i)29-s + (−0.981 + 0.189i)31-s + (0.995 − 0.0950i)33-s + (0.786 − 0.618i)37-s + (0.995 + 0.0950i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.945 - 0.325i$
Analytic conductor: \(2.99072\)
Root analytic conductor: \(2.99072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 644,\ (0:\ ),\ 0.945 - 0.325i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100176267 - 0.1838332238i\)
\(L(\frac12)\) \(\approx\) \(1.100176267 - 0.1838332238i\)
\(L(1)\) \(\approx\) \(0.9671388141 + 0.1213678055i\)
\(L(1)\) \(\approx\) \(0.9671388141 + 0.1213678055i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.327 + 0.945i)T \)
5 \( 1 + (-0.928 + 0.371i)T \)
11 \( 1 + (0.235 - 0.971i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (-0.0475 - 0.998i)T \)
19 \( 1 + (0.0475 - 0.998i)T \)
29 \( 1 + (0.841 - 0.540i)T \)
31 \( 1 + (-0.981 + 0.189i)T \)
37 \( 1 + (0.786 - 0.618i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.995 + 0.0950i)T \)
59 \( 1 + (-0.580 - 0.814i)T \)
61 \( 1 + (0.327 - 0.945i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (-0.888 + 0.458i)T \)
79 \( 1 + (-0.995 + 0.0950i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
89 \( 1 + (-0.981 - 0.189i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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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

−23.363160294591764770656180039255, −22.28730361367253224396099098007, −21.08009102153094104009162177946, −20.212756790307606910419403969670, −19.70502710722612616767071864535, −18.83387618015418121114283612814, −18.23439400291264388693072069144, −17.10915892643558648071639464644, −16.45871908079174317507029393492, −15.25309780030414891567322738329, −14.65618429777298082569539431199, −13.68773359354378933028029559201, −12.67722455786572538699766129994, −12.16114690986329334214565708371, −11.42611704208627729991296894565, −10.21031111738151914340626550060, −8.92805800212627781552798158970, −8.385750364595215323281280147974, −7.38609207401963654071232689547, −6.76251159192694868474310862339, −5.62629909344668507026455066862, −4.25618157485285248056089589016, −3.56623068201761222782204793493, −2.09956426807208255009586872804, −1.23904731437230770421265628672, 0.59324740789683884500760418246, 2.74867667526335375337799349450, 3.28124098150771756298808005860, 4.28449054682671153720971866669, 5.19837658629344729714571860418, 6.32317294302283043421693494883, 7.56222318551231134713639279602, 8.345025036155954032346399930737, 9.137638625757824697825036735729, 10.16734131566553541551656298888, 11.25304772171246152934219183485, 11.35645719411374558353052647009, 12.84556888737868839730003521967, 13.87823973595833016816765673829, 14.604085788808235052694036457219, 15.58584960091331432938493510394, 15.931408005385442210618907838502, 16.81258003136148774853577926862, 17.9813519655863863602220992199, 18.84010157948966099115822902360, 19.93374537438046682741946064600, 20.08053421457279001562291256284, 21.346164130594946274574393671878, 21.95671052897626480205031464631, 22.817760751704119857032849183966

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