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Properties

Label 2-1407-7.4-c1-0-1
Degree $2$
Conductor $1407$
Sign $-0.966 + 0.256i$
Analytic cond. $11.2349$
Root an. cond. $3.35185$
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  + (0.737 + 1.27i)2-s + (−0.5 + 0.866i)3-s + (−0.0884 + 0.153i)4-s + (−1.74 − 3.02i)5-s − 1.47·6-s + (−0.839 − 2.50i)7-s + 2.68·8-s + (−0.499 − 0.866i)9-s + (2.57 − 4.46i)10-s + (−2.73 + 4.73i)11-s + (−0.0884 − 0.153i)12-s − 6.33·13-s + (2.58 − 2.92i)14-s + 3.49·15-s + (2.16 + 3.74i)16-s + (2.06 − 3.58i)17-s + ⋯
L(s)  = 1  + (0.521 + 0.903i)2-s + (−0.288 + 0.499i)3-s + (−0.0442 + 0.0766i)4-s + (−0.781 − 1.35i)5-s − 0.602·6-s + (−0.317 − 0.948i)7-s + 0.951·8-s + (−0.166 − 0.288i)9-s + (0.815 − 1.41i)10-s + (−0.824 + 1.42i)11-s + (−0.0255 − 0.0442i)12-s − 1.75·13-s + (0.691 − 0.781i)14-s + 0.902·15-s + (0.540 + 0.935i)16-s + (0.501 − 0.868i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.966 + 0.256i$
Analytic conductor: \(11.2349\)
Root analytic conductor: \(3.35185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :1/2),\ -0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2717085615\)
\(L(\frac12)\) \(\approx\) \(0.2717085615\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.839 + 2.50i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.737 - 1.27i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.74 + 3.02i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.73 - 4.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.33T + 13T^{2} \)
17 \( 1 + (-2.06 + 3.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.57 - 6.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.06 - 3.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.01T + 29T^{2} \)
31 \( 1 + (3.28 - 5.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.628 - 1.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 3.99T + 43T^{2} \)
47 \( 1 + (4.30 + 7.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.30 - 2.25i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.69 - 4.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.01 + 5.22i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + 9.04T + 71T^{2} \)
73 \( 1 + (-1.47 + 2.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.60 - 6.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.42T + 83T^{2} \)
89 \( 1 + (-1.97 - 3.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.40T + 97T^{2} \)
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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

−9.894343142373508507407831692387, −9.388762051627575271161933968772, −7.88755286814062390316224375736, −7.49673704184425035291570396307, −6.99168002226546841366347430297, −5.33213889619879551331092783818, −5.13863025303489495487324667390, −4.47196275828650077410977879142, −3.51880922326914239709666972956, −1.55677719410276874883707920394, 0.091020929869261881994020259503, 2.26645643219107483574088920213, 2.90197440124866594064504191491, 3.43157934723474484120666588158, 4.85215428920899721360384554356, 5.70330004272211334763555200660, 6.74912799914885386872385957974, 7.53079925752624297423262419585, 8.003141979599279745363827900456, 9.274039531792388571743524390929

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