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Properties

Label 1-1449-1449.661-r0-0-0
Degree $1$
Conductor $1449$
Sign $0.0499 - 0.998i$
Analytic cond. $6.72912$
Root an. cond. $6.72912$
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)2-s + (−0.786 − 0.618i)4-s + (0.415 − 0.909i)5-s + (0.841 − 0.540i)8-s + (0.723 + 0.690i)10-s + (0.654 − 0.755i)11-s + (−0.723 − 0.690i)13-s + (0.235 + 0.971i)16-s + (−0.786 + 0.618i)17-s + (−0.786 − 0.618i)19-s + (−0.888 + 0.458i)20-s + (0.5 + 0.866i)22-s + (−0.654 − 0.755i)25-s + (0.888 − 0.458i)26-s + (0.928 + 0.371i)29-s + ⋯
L(s)  = 1  + (−0.327 + 0.945i)2-s + (−0.786 − 0.618i)4-s + (0.415 − 0.909i)5-s + (0.841 − 0.540i)8-s + (0.723 + 0.690i)10-s + (0.654 − 0.755i)11-s + (−0.723 − 0.690i)13-s + (0.235 + 0.971i)16-s + (−0.786 + 0.618i)17-s + (−0.786 − 0.618i)19-s + (−0.888 + 0.458i)20-s + (0.5 + 0.866i)22-s + (−0.654 − 0.755i)25-s + (0.888 − 0.458i)26-s + (0.928 + 0.371i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0499 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0499 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $0.0499 - 0.998i$
Analytic conductor: \(6.72912\)
Root analytic conductor: \(6.72912\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (661, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1449,\ (0:\ ),\ 0.0499 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6232852199 - 0.5928743224i\)
\(L(\frac12)\) \(\approx\) \(0.6232852199 - 0.5928743224i\)
\(L(1)\) \(\approx\) \(0.8157389330 + 0.01064274733i\)
\(L(1)\) \(\approx\) \(0.8157389330 + 0.01064274733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.327 + 0.945i)T \)
5 \( 1 + (0.415 - 0.909i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (-0.723 - 0.690i)T \)
17 \( 1 + (-0.786 + 0.618i)T \)
19 \( 1 + (-0.786 - 0.618i)T \)
29 \( 1 + (0.928 + 0.371i)T \)
31 \( 1 + (-0.0475 - 0.998i)T \)
37 \( 1 + (0.995 + 0.0950i)T \)
41 \( 1 + (0.995 - 0.0950i)T \)
43 \( 1 + (0.888 - 0.458i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.235 - 0.971i)T \)
59 \( 1 + (-0.235 + 0.971i)T \)
61 \( 1 + (-0.888 - 0.458i)T \)
67 \( 1 + (-0.981 + 0.189i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.928 + 0.371i)T \)
79 \( 1 + (-0.723 - 0.690i)T \)
83 \( 1 + (-0.995 - 0.0950i)T \)
89 \( 1 + (-0.888 + 0.458i)T \)
97 \( 1 + (0.580 + 0.814i)T \)
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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

−20.99333401738153316045462078625, −19.92514658228400595239065470301, −19.49955972661593613543308217128, −18.68863040362294437652716475038, −17.968391682995888355219142301343, −17.39922044313500434825779395420, −16.70418839025642699697761157077, −15.55872348826243901800003592839, −14.4114407176505664355690209863, −14.22121280464844947600575741414, −13.16124490998587298077441825189, −12.31367398573850714556063874762, −11.66038898449346660130981522446, −10.85861555504683007711542935182, −10.13438518900059680394569638263, −9.46383588217879168678408777479, −8.80243745766897203872009483150, −7.58142870284132446244751912642, −6.93940172362216248292351099499, −6.00521854575286293717918832170, −4.61506034858703454475728582934, −4.14089766011753234051998749813, −2.869808341189557392830960654, −2.291569799616497985287414852, −1.4032898378469636753305134317, 0.38102884957661617452877847527, 1.37788847216243925578288977764, 2.621183033919195237118518322578, 4.19679129199473920522198137755, 4.63585958017480992273417387011, 5.81855423511814409020747354000, 6.15176489991012918630195480079, 7.26389957394802785395478068512, 8.173203122243869771966369403117, 8.83050762043678308283168908428, 9.38936377287493670585990808189, 10.319323447565064941758273685975, 11.15161230901695884250477927578, 12.4120174666376159072062612043, 13.05864747745840760686607519333, 13.739187643969897277812580787865, 14.60115423928239063110073762274, 15.30743851586745877189445963603, 16.13820437366910826143548762847, 16.8054978837022598182484999646, 17.467417315895959820356001698829, 17.85032836503847278182854719482, 19.13962323939850658111413034275, 19.582079400577625061668898247965, 20.34645510044297024961716797419

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