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Properties

Label 2-1372-7.4-c1-0-17
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  + (−1.59 + 2.75i)3-s + (0.598 + 1.03i)5-s + (−3.57 − 6.18i)9-s + (1.07 − 1.85i)11-s + 4.07·13-s − 3.81·15-s + (1.29 − 2.24i)17-s + (−2.11 − 3.66i)19-s + (−4.49 − 7.78i)23-s + (1.78 − 3.08i)25-s + 13.2·27-s + 6.30·29-s + (2.76 − 4.78i)31-s + (3.41 + 5.90i)33-s + (−1.31 − 2.28i)37-s + ⋯
L(s)  = 1  + (−0.919 + 1.59i)3-s + (0.267 + 0.463i)5-s + (−1.19 − 2.06i)9-s + (0.322 − 0.559i)11-s + 1.13·13-s − 0.984·15-s + (0.314 − 0.544i)17-s + (−0.485 − 0.840i)19-s + (−0.937 − 1.62i)23-s + (0.356 − 0.617i)25-s + 2.54·27-s + 1.17·29-s + (0.496 − 0.859i)31-s + (0.593 + 1.02i)33-s + (−0.216 − 0.374i)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: $\chi_{1372} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1372,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.046934509\)
\(L(\frac12)\) \(\approx\) \(1.046934509\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 2.75i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.598 - 1.03i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.07 + 1.85i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.07T + 13T^{2} \)
17 \( 1 + (-1.29 + 2.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.11 + 3.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.49 + 7.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.30T + 29T^{2} \)
31 \( 1 + (-2.76 + 4.78i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.31 + 2.28i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.34T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + (5.52 + 9.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.47 + 2.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.74 - 8.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.03 - 6.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.25 + 10.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.149T + 71T^{2} \)
73 \( 1 + (4.62 - 8.00i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.02 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.15T + 83T^{2} \)
89 \( 1 + (6.59 + 11.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.54T + 97T^{2} \)
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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

−9.989823858061802899277051171612, −8.801231377691827814916303609242, −8.414251394990721567032556721578, −6.52537941000576074419038429749, −6.38633849279387087649656337016, −5.32569012107414751462997338019, −4.49084798443306485869193918575, −3.73414513477433166552825210122, −2.71497009244207444066513123382, −0.52981578972420275060035859104, 1.33934452137518379913313479541, 1.70234406562058765541281098284, 3.40113141569518583604804465944, 4.80850093077135611848225478814, 5.69680389656224253520104079825, 6.32132002820387101506638824415, 6.95194880814271113023930850696, 8.048579506292342626077792658152, 8.357566093060223820578199427785, 9.631272597819102227240346082930

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