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Properties

Label 2-1372-7.2-c1-0-11
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  + (0.369 + 0.640i)3-s + (−0.975 + 1.68i)5-s + (1.22 − 2.12i)9-s + (−2.24 − 3.89i)11-s + 5.82·13-s − 1.44·15-s + (0.907 + 1.57i)17-s + (1.90 − 3.30i)19-s + (−1.17 + 2.03i)23-s + (0.598 + 1.03i)25-s + 4.03·27-s − 5.66·29-s + (−4.69 − 8.13i)31-s + (1.66 − 2.88i)33-s + (4.00 − 6.94i)37-s + ⋯
L(s)  = 1  + (0.213 + 0.369i)3-s + (−0.436 + 0.755i)5-s + (0.408 − 0.707i)9-s + (−0.678 − 1.17i)11-s + 1.61·13-s − 0.372·15-s + (0.220 + 0.381i)17-s + (0.437 − 0.757i)19-s + (−0.245 + 0.424i)23-s + (0.119 + 0.207i)25-s + 0.776·27-s − 1.05·29-s + (−0.843 − 1.46i)31-s + (0.289 − 0.501i)33-s + (0.659 − 1.14i)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

\(L(1)\) \(\approx\) \(1.761825622\)
\(L(\frac12)\) \(\approx\) \(1.761825622\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.369 - 0.640i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.975 - 1.68i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.24 + 3.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.82T + 13T^{2} \)
17 \( 1 + (-0.907 - 1.57i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.90 + 3.30i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.17 - 2.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.66T + 29T^{2} \)
31 \( 1 + (4.69 + 8.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.00 + 6.94i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 - 9.11T + 43T^{2} \)
47 \( 1 + (1.98 - 3.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.65 - 4.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.84 - 3.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.90 - 6.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.930 - 1.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.149T + 71T^{2} \)
73 \( 1 + (-1.40 - 2.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.43 - 2.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + (-0.530 + 0.919i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.97T + 97T^{2} \)
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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

−9.346108224030106557831942426247, −9.008744864533197974640196608398, −7.85533572161702435995705226651, −7.33488722958484680540742812920, −6.08102845338048522312695492408, −5.72552006842142599094224993302, −4.05274985808536826090862788311, −3.63491148122160438397487624899, −2.67772565219208429423546563155, −0.884979284003021302754440636152, 1.15053557834764869852576539230, 2.19473886402232209356740364659, 3.58291854611212774298058892746, 4.54131608921353922816913043577, 5.27363153113257858054437415833, 6.34091292430604405968664081099, 7.44873078230327956775697410231, 7.88179248892995502035995418322, 8.655214376727077662069055460446, 9.495957721429572908546563624425

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