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Properties

Label 1-449-449.164-r0-0-0
Degree $1$
Conductor $449$
Sign $0.995 - 0.0920i$
Analytic cond. $2.08514$
Root an. cond. $2.08514$
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  + (−0.578 + 0.815i)2-s + (0.764 + 0.645i)3-s + (−0.330 − 0.943i)4-s + (−0.222 − 0.974i)5-s + (−0.968 + 0.249i)6-s + (0.846 − 0.532i)7-s + (0.960 + 0.276i)8-s + (0.167 + 0.985i)9-s + (0.923 + 0.382i)10-s + (−0.974 + 0.222i)11-s + (0.356 − 0.934i)12-s + (0.508 − 0.861i)13-s + (−0.0560 + 0.998i)14-s + (0.458 − 0.888i)15-s + (−0.781 + 0.623i)16-s + (−0.139 − 0.990i)17-s + ⋯
L(s)  = 1  + (−0.578 + 0.815i)2-s + (0.764 + 0.645i)3-s + (−0.330 − 0.943i)4-s + (−0.222 − 0.974i)5-s + (−0.968 + 0.249i)6-s + (0.846 − 0.532i)7-s + (0.960 + 0.276i)8-s + (0.167 + 0.985i)9-s + (0.923 + 0.382i)10-s + (−0.974 + 0.222i)11-s + (0.356 − 0.934i)12-s + (0.508 − 0.861i)13-s + (−0.0560 + 0.998i)14-s + (0.458 − 0.888i)15-s + (−0.781 + 0.623i)16-s + (−0.139 − 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $0.995 - 0.0920i$
Analytic conductor: \(2.08514\)
Root analytic conductor: \(2.08514\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (0:\ ),\ 0.995 - 0.0920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.199646683 - 0.05532571774i\)
\(L(\frac12)\) \(\approx\) \(1.199646683 - 0.05532571774i\)
\(L(1)\) \(\approx\) \(0.9917328442 + 0.1733757222i\)
\(L(1)\) \(\approx\) \(0.9917328442 + 0.1733757222i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (-0.578 + 0.815i)T \)
3 \( 1 + (0.764 + 0.645i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (0.846 - 0.532i)T \)
11 \( 1 + (-0.974 + 0.222i)T \)
13 \( 1 + (0.508 - 0.861i)T \)
17 \( 1 + (-0.139 - 0.990i)T \)
19 \( 1 + (0.0280 - 0.999i)T \)
23 \( 1 + (0.943 + 0.330i)T \)
29 \( 1 + (-0.912 + 0.408i)T \)
31 \( 1 + (-0.912 - 0.408i)T \)
37 \( 1 + (0.831 - 0.555i)T \)
41 \( 1 + (0.745 - 0.666i)T \)
43 \( 1 + (-0.356 - 0.934i)T \)
47 \( 1 + (0.968 - 0.249i)T \)
53 \( 1 + (-0.745 + 0.666i)T \)
59 \( 1 + (0.998 + 0.0560i)T \)
61 \( 1 + (0.815 - 0.578i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.831 - 0.555i)T \)
73 \( 1 + (0.999 + 0.0280i)T \)
79 \( 1 + (0.980 - 0.195i)T \)
83 \( 1 + (-0.645 + 0.764i)T \)
89 \( 1 + (0.993 - 0.111i)T \)
97 \( 1 + (0.276 + 0.960i)T \)
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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

−23.95180324474718192842806308496, −23.24578524655002188397723251765, −21.99833452188957327765049031484, −21.137024114471938106620189561228, −20.69727427007387746771115584330, −19.45504272744131971001002382094, −18.76689149604532109248836436977, −18.40231882089542706111684754865, −17.6252380812936854222249034096, −16.32202977593925344711720298751, −15.03475586127415237557785099662, −14.41718034658484951376751248558, −13.35224122740333254792684089459, −12.563657238540457380106940137147, −11.45882201115219230464936175306, −10.90591165388985649496544827636, −9.767538990696248512610012267271, −8.6559081499312769243474101903, −8.028201125615516887796785909812, −7.257154326889514149317717838625, −6.00978381146534490220701047098, −4.24161214482212973713446549660, −3.24241191901569571529710648294, −2.316127836770448894080638357042, −1.51647499210084626845040174555, 0.79605037294486889560482209383, 2.24489108578033639373253786103, 3.902666454755788264392139666658, 5.03806979273046090735550486569, 5.31744655939212033473043980266, 7.42872443312409843138881093255, 7.73171334048094205219853811707, 8.827259023836853266083342718019, 9.32325649079707936563747387719, 10.56361637379065932946676807518, 11.17130724701339096197766029889, 13.09354485426179279721369318795, 13.545511086666639537254366470590, 14.704665683425403349710370418119, 15.47489194668179098195269368900, 16.04900988047591821292874217754, 16.96085884315350935651359858178, 17.81329991168251240487353677962, 18.73213855939001607754520190875, 19.96779689407743767170053098720, 20.389974785021485755926912501415, 21.05887040904424300135756983986, 22.4100434231434037166518184105, 23.53341206582502598101152952787, 24.05879281329943111422362947962

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