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Properties

Label 1-644-644.471-r0-0-0
Degree $1$
Conductor $644$
Sign $-0.568 + 0.822i$
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  + (−0.723 + 0.690i)3-s + (0.888 + 0.458i)5-s + (0.0475 − 0.998i)9-s + (−0.995 − 0.0950i)11-s + (−0.142 − 0.989i)13-s + (−0.959 + 0.281i)15-s + (0.327 + 0.945i)17-s + (−0.327 + 0.945i)19-s + (0.580 + 0.814i)25-s + (0.654 + 0.755i)27-s + (−0.654 + 0.755i)29-s + (−0.235 + 0.971i)31-s + (0.786 − 0.618i)33-s + (−0.0475 + 0.998i)37-s + (0.786 + 0.618i)39-s + ⋯
L(s)  = 1  + (−0.723 + 0.690i)3-s + (0.888 + 0.458i)5-s + (0.0475 − 0.998i)9-s + (−0.995 − 0.0950i)11-s + (−0.142 − 0.989i)13-s + (−0.959 + 0.281i)15-s + (0.327 + 0.945i)17-s + (−0.327 + 0.945i)19-s + (0.580 + 0.814i)25-s + (0.654 + 0.755i)27-s + (−0.654 + 0.755i)29-s + (−0.235 + 0.971i)31-s + (0.786 − 0.618i)33-s + (−0.0475 + 0.998i)37-s + (0.786 + 0.618i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.568 + 0.822i$
Analytic conductor: \(2.99072\)
Root analytic conductor: \(2.99072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 644,\ (0:\ ),\ -0.568 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4218388176 + 0.8044946262i\)
\(L(\frac12)\) \(\approx\) \(0.4218388176 + 0.8044946262i\)
\(L(1)\) \(\approx\) \(0.7869240473 + 0.3452173086i\)
\(L(1)\) \(\approx\) \(0.7869240473 + 0.3452173086i\)

Euler product

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

−22.72797601856192613457298632757, −21.6836627093178082721129675492, −21.1819728106936205915098397587, −20.17249588459415359262069887341, −19.12273936023373526118493022220, −18.29554659850274910030186051164, −17.794949478962892710427742894452, −16.75020509526589285402287439476, −16.38728204211439037764807897723, −15.160634810453894402931212403166, −13.85602508431868350621296953325, −13.39379026721904997608127508928, −12.61432479641061229689773554706, −11.67810038717855207038299160671, −10.87470220567253297815837553481, −9.83913687297618004974557590344, −9.031527490698827718879205844111, −7.8089164514486419971400415809, −6.99245308651391447429189774752, −6.024213104639648156507130126510, −5.24260864237337337410903988571, −4.46305440291825564189982359961, −2.56894870984751686791788499806, −1.890366942721152975096543322691, −0.49216704480374946542663831858, 1.41546913712394596432778276455, 2.813982315640496730965201948494, 3.724827697195009806850333006353, 5.12182604164750800869333392493, 5.65192092081194662295305138185, 6.46588769965943149507277200126, 7.6759264621910783017719000635, 8.78830190929364583459445451674, 9.952553366901705522025155028, 10.439361934091795763283683098426, 10.98992474478993800903702181153, 12.39832071331860651244465430013, 12.92391223533939084639509281093, 14.14627340349900434100469850697, 14.96844564584212391139140629372, 15.66852498658806089768246849957, 16.72520046178592727497377988803, 17.31735184049024824246502190279, 18.16597982795028919743241169813, 18.745127518045534813880203664019, 20.16267647791911514929140426433, 20.94759014552425616749568501166, 21.585572529715286332598897008229, 22.25530210798350406518914062086, 23.08380871661473119115818265596

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