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Properties

Label 1-644-644.363-r0-0-0
Degree $1$
Conductor $644$
Sign $0.899 + 0.436i$
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  + (−0.142 − 0.989i)3-s + (0.959 + 0.281i)5-s + (−0.959 + 0.281i)9-s + (−0.841 + 0.540i)11-s + (0.654 + 0.755i)13-s + (0.142 − 0.989i)15-s + (−0.415 + 0.909i)17-s + (0.415 + 0.909i)19-s + (0.841 + 0.540i)25-s + (0.415 + 0.909i)27-s + (0.415 − 0.909i)29-s + (−0.142 + 0.989i)31-s + (0.654 + 0.755i)33-s + (−0.959 + 0.281i)37-s + (0.654 − 0.755i)39-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)3-s + (0.959 + 0.281i)5-s + (−0.959 + 0.281i)9-s + (−0.841 + 0.540i)11-s + (0.654 + 0.755i)13-s + (0.142 − 0.989i)15-s + (−0.415 + 0.909i)17-s + (0.415 + 0.909i)19-s + (0.841 + 0.540i)25-s + (0.415 + 0.909i)27-s + (0.415 − 0.909i)29-s + (−0.142 + 0.989i)31-s + (0.654 + 0.755i)33-s + (−0.959 + 0.281i)37-s + (0.654 − 0.755i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.899 + 0.436i$
Analytic conductor: \(2.99072\)
Root analytic conductor: \(2.99072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 644,\ (0:\ ),\ 0.899 + 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.341960710 + 0.3081797666i\)
\(L(\frac12)\) \(\approx\) \(1.341960710 + 0.3081797666i\)
\(L(1)\) \(\approx\) \(1.110513721 - 0.04327736604i\)
\(L(1)\) \(\approx\) \(1.110513721 - 0.04327736604i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.142 - 0.989i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
11 \( 1 + (-0.841 + 0.540i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (-0.959 + 0.281i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.654 + 0.755i)T \)
59 \( 1 + (-0.654 - 0.755i)T \)
61 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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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

−22.495401466244861941087345889223, −22.07375851282107609824667569622, −21.043413036668013197027282339196, −20.68534954591532533672498718544, −19.85334094046235998683083579169, −18.46766609500299880581173088592, −17.7920881345614872622269323652, −17.06232633970948691343466947475, −15.93240621538407557761475641779, −15.742482550609490512574226960778, −14.46025728967366144981968943650, −13.64026834896978626037234147444, −12.98333243398730635116591354415, −11.683492362265213889376442533689, −10.73254204271989414527708158554, −10.23691968307066005647658710951, −9.11168974484156308047443437316, −8.686896543942997289092056685788, −7.30478495962723896261857091186, −5.946841593457336632557010646787, −5.41050709002911989977889636223, −4.56583172283344719556312855098, −3.228447485587920563422886295303, −2.4567646048622125157079416542, −0.709527585331183822546938531834, 1.42946247293459498370219802174, 2.0918386648414876252629276656, 3.18682703571238720559407630413, 4.7018746928641524461155656527, 5.869900371321709329543778418280, 6.362878609399805386225461312720, 7.37771085044491203475498242915, 8.27997687747112956750796473062, 9.26788443390714745563189825102, 10.35686044489170725923126245035, 11.06335282462660786995084051705, 12.24420742620185356706193826254, 12.8943911081085810327060744699, 13.77944614324599122804361302465, 14.26653241790245133146123166873, 15.43917741347769961688373069080, 16.534708650769323247009224731107, 17.425601751632940377382625313503, 18.01596518558847726333907632622, 18.69729645860016458996692155455, 19.46097928249490268782393685071, 20.61080731742552817374552426694, 21.22834887603596743704415660954, 22.21858250645729463959350762011, 23.10300062974419871667837341518

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