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Properties

Label 1-421-421.48-r0-0-0
Degree $1$
Conductor $421$
Sign $0.955 + 0.293i$
Analytic cond. $1.95511$
Root an. cond. $1.95511$
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  + (0.983 − 0.178i)2-s + (−0.963 + 0.266i)3-s + (0.936 − 0.351i)4-s + (0.936 + 0.351i)5-s + (−0.900 + 0.433i)6-s + (−0.963 − 0.266i)7-s + (0.858 − 0.512i)8-s + (0.858 − 0.512i)9-s + (0.983 + 0.178i)10-s + (−0.0448 + 0.998i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (−0.995 − 0.0896i)14-s + (−0.995 − 0.0896i)15-s + (0.753 − 0.657i)16-s + (0.473 − 0.880i)17-s + ⋯
L(s)  = 1  + (0.983 − 0.178i)2-s + (−0.963 + 0.266i)3-s + (0.936 − 0.351i)4-s + (0.936 + 0.351i)5-s + (−0.900 + 0.433i)6-s + (−0.963 − 0.266i)7-s + (0.858 − 0.512i)8-s + (0.858 − 0.512i)9-s + (0.983 + 0.178i)10-s + (−0.0448 + 0.998i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (−0.995 − 0.0896i)14-s + (−0.995 − 0.0896i)15-s + (0.753 − 0.657i)16-s + (0.473 − 0.880i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $0.955 + 0.293i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ 0.955 + 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.032998893 + 0.3054224523i\)
\(L(\frac12)\) \(\approx\) \(2.032998893 + 0.3054224523i\)
\(L(1)\) \(\approx\) \(1.605920920 + 0.08692474007i\)
\(L(1)\) \(\approx\) \(1.605920920 + 0.08692474007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad421 \( 1 \)
good2 \( 1 + (0.983 - 0.178i)T \)
3 \( 1 + (-0.963 + 0.266i)T \)
5 \( 1 + (0.936 + 0.351i)T \)
7 \( 1 + (-0.963 - 0.266i)T \)
11 \( 1 + (-0.0448 + 0.998i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.473 - 0.880i)T \)
19 \( 1 + (0.134 + 0.990i)T \)
23 \( 1 + (-0.550 - 0.834i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.393 - 0.919i)T \)
37 \( 1 + (-0.0448 + 0.998i)T \)
41 \( 1 + (0.134 + 0.990i)T \)
43 \( 1 + (0.753 - 0.657i)T \)
47 \( 1 + (0.936 + 0.351i)T \)
53 \( 1 + (-0.963 - 0.266i)T \)
59 \( 1 + (-0.691 + 0.722i)T \)
61 \( 1 + (0.753 - 0.657i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.691 - 0.722i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.753 + 0.657i)T \)
83 \( 1 + (-0.691 + 0.722i)T \)
89 \( 1 + (-0.393 + 0.919i)T \)
97 \( 1 + (0.134 - 0.990i)T \)
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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

−23.96064740383062602309312440761, −23.365311310977432154980924814394, −22.31861477804749498432776499987, −21.79702111834694204995045984870, −21.23963914200804707349803317, −19.91860010239015818400980418982, −19.05930331893968054151054339511, −17.73899220022650748706155744590, −17.17898950691766533396269574119, −15.987269839207951550449560735306, −15.848532049891321241864576189791, −14.23248925013225260210971003574, −13.311240739850280843869878796259, −12.83416399783813177244618746405, −12.04652350348672167747752394063, −10.835102679254217911292480160390, −10.21354921340515463482986047312, −8.78952140880866850754505479996, −7.44098629275194329401298141299, −6.25817543308427755654708459521, −5.851579327567028499817156125883, −5.13091736912407091736760565623, −3.68729225596672495188286080550, −2.56847635594705352755439699639, −1.133069475498015663808140790087, 1.41411295179472240023425649455, 2.651848141267166290061577786627, 3.93826917419236596321443202030, 4.82831476816651888674255085398, 5.95802127471413657215828832431, 6.495969029958991367632497305069, 7.30782168981809766146717658623, 9.62637622556235945959968280958, 9.97873998183522964672370441173, 10.915033611775379002457727569740, 12.04681436526574423107198119058, 12.61505693616761271693148859727, 13.65904340483178447738918054893, 14.40095250915002435822075937238, 15.53656525149547613552473902838, 16.41203925617115726798384055388, 16.95763665802118154648380551079, 18.2576226925381744195313966686, 18.94455247651209376981262972324, 20.417382497015361338947432385840, 20.93512401375553804413807303918, 21.96353382683516435604498128515, 22.52581784266106333550041935469, 23.09250488673075960149879650797, 23.92528944745067710657592203705

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