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Properties

Label 2-1421-29.28-c1-0-5
Degree $2$
Conductor $1421$
Sign $0.945 + 0.326i$
Analytic cond. $11.3467$
Root an. cond. $3.36849$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.52i·2-s + 3.09i·3-s − 0.332·4-s − 3.78·5-s − 4.72·6-s + 2.54i·8-s − 6.58·9-s − 5.77i·10-s − 1.07i·11-s − 1.02i·12-s − 2.93·13-s − 11.7i·15-s − 4.55·16-s + 5.85i·17-s − 10.0i·18-s − 4.24i·19-s + ⋯
L(s)  = 1  + 1.07i·2-s + 1.78i·3-s − 0.166·4-s − 1.69·5-s − 1.93·6-s + 0.900i·8-s − 2.19·9-s − 1.82i·10-s − 0.322i·11-s − 0.296i·12-s − 0.815·13-s − 3.02i·15-s − 1.13·16-s + 1.41i·17-s − 2.37i·18-s − 0.974i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1421\)    =    \(7^{2} \cdot 29\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(11.3467\)
Root analytic conductor: \(3.36849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1421} (1275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1421,\ (\ :1/2),\ 0.945 + 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3619725027\)
\(L(\frac12)\) \(\approx\) \(0.3619725027\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 + (5.08 + 1.75i)T \)
good2 \( 1 - 1.52iT - 2T^{2} \)
3 \( 1 - 3.09iT - 3T^{2} \)
5 \( 1 + 3.78T + 5T^{2} \)
11 \( 1 + 1.07iT - 11T^{2} \)
13 \( 1 + 2.93T + 13T^{2} \)
17 \( 1 - 5.85iT - 17T^{2} \)
19 \( 1 + 4.24iT - 19T^{2} \)
23 \( 1 - 7.62T + 23T^{2} \)
31 \( 1 - 4.39iT - 31T^{2} \)
37 \( 1 - 5.29iT - 37T^{2} \)
41 \( 1 + 0.682iT - 41T^{2} \)
43 \( 1 + 1.96iT - 43T^{2} \)
47 \( 1 - 5.06iT - 47T^{2} \)
53 \( 1 - 8.54T + 53T^{2} \)
59 \( 1 + 4.38T + 59T^{2} \)
61 \( 1 + 7.36iT - 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 4.52T + 71T^{2} \)
73 \( 1 + 7.69iT - 73T^{2} \)
79 \( 1 + 16.8iT - 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + 3.79iT - 89T^{2} \)
97 \( 1 - 8.72iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.45824979317829073229929009220, −9.119474470345793396208580009063, −8.732242778477934145604467958438, −7.900604141736223473126964101262, −7.20618233253140351316716595189, −6.17959049132937842478331723785, −5.04111399223280862484343430769, −4.66613812973394734983580065714, −3.69077266016760933387865720401, −2.90452305313197896322310244690, 0.16135009064389404927651943289, 1.14885761266780642846866594338, 2.40747544125513573520578510040, 3.11474175875587959942291957966, 4.16812223098583548976277906066, 5.41466718735243181781263318755, 6.82904253467647849220018253649, 7.29254369846046099614823642890, 7.64403501658309009292167764817, 8.686977738050262664458785721711

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