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Properties

Label 2-1372-7.2-c1-0-6
Degree $2$
Conductor $1372$
Sign $1$
Analytic cond. $10.9554$
Root an. cond. $3.30990$
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  + (−0.881 − 1.52i)3-s + (−1.02 + 1.76i)5-s + (−0.0549 + 0.0951i)9-s + (−0.0990 − 0.171i)11-s + 1.13·13-s + 3.60·15-s + (−0.489 − 0.847i)17-s + (−3.00 + 5.20i)19-s + (0.0244 − 0.0423i)23-s + (0.411 + 0.713i)25-s − 5.09·27-s + 7.43·29-s + (−0.236 − 0.410i)31-s + (−0.174 + 0.302i)33-s + (0.592 − 1.02i)37-s + ⋯
L(s)  = 1  + (−0.509 − 0.881i)3-s + (−0.456 + 0.791i)5-s + (−0.0183 + 0.0317i)9-s + (−0.0298 − 0.0517i)11-s + 0.314·13-s + 0.930·15-s + (−0.118 − 0.205i)17-s + (−0.688 + 1.19i)19-s + (0.00509 − 0.00883i)23-s + (0.0823 + 0.142i)25-s − 0.980·27-s + 1.38·29-s + (−0.0425 − 0.0737i)31-s + (−0.0304 + 0.0526i)33-s + (0.0974 − 0.168i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1372\)    =    \(2^{2} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(10.9554\)
Root analytic conductor: \(3.30990\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1372} (1353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1372,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.196164653\)
\(L(\frac12)\) \(\approx\) \(1.196164653\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.881 + 1.52i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.02 - 1.76i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.0990 + 0.171i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.13T + 13T^{2} \)
17 \( 1 + (0.489 + 0.847i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.00 - 5.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0244 + 0.0423i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.43T + 29T^{2} \)
31 \( 1 + (0.236 + 0.410i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.592 + 1.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.38T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + (2.31 - 4.00i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.22 - 3.85i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.67 - 11.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.36 + 11.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.66 + 6.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.14T + 71T^{2} \)
73 \( 1 + (-5.08 - 8.81i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.22 + 5.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + (-5.45 + 9.44i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−9.663680008509072758528226674090, −8.616962567259798252607376588467, −7.74082613038850196924777781988, −7.13625413955436473623615477620, −6.33650269296186183767942133795, −5.78804361784014483593005939028, −4.39992262155536812083969815533, −3.46746435335428377359011010750, −2.29286887356423998469169203806, −0.944232757741230858676435735817, 0.71565119774083214016255450073, 2.43052123990835903121406438028, 3.88097475767241181034533046700, 4.54395634482743993592455412742, 5.14174039874756494093274259833, 6.15842364711419592522521922488, 7.12328573732646722486902172506, 8.209229734200229923388888689223, 8.781297540332856754389181568813, 9.623551473980549534549084580312

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