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Properties

Label 2-1372-7.2-c1-0-4
Degree $2$
Conductor $1372$
Sign $-0.5 - 0.866i$
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  + (0.653 + 1.13i)3-s + (−0.483 + 0.837i)5-s + (0.646 − 1.11i)9-s + (1.89 + 3.27i)11-s − 2.23·13-s − 1.26·15-s + (−1.22 − 2.12i)17-s + (−4.11 + 7.13i)19-s + (−2.43 + 4.22i)23-s + (2.03 + 3.52i)25-s + 5.60·27-s + 2.43·29-s + (−0.826 − 1.43i)31-s + (−2.47 + 4.28i)33-s + (1.82 − 3.15i)37-s + ⋯
L(s)  = 1  + (0.377 + 0.653i)3-s + (−0.216 + 0.374i)5-s + (0.215 − 0.372i)9-s + (0.570 + 0.988i)11-s − 0.619·13-s − 0.326·15-s + (−0.297 − 0.514i)17-s + (−0.945 + 1.63i)19-s + (−0.508 + 0.880i)23-s + (0.406 + 0.704i)25-s + 1.07·27-s + 0.452·29-s + (−0.148 − 0.257i)31-s + (−0.430 + 0.746i)33-s + (0.299 − 0.519i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1372\)    =    \(2^{2} \cdot 7^{3}\)
Sign: $-0.5 - 0.866i$
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: \((2,\ 1372,\ (\ :1/2),\ -0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.512925911\)
\(L(\frac12)\) \(\approx\) \(1.512925911\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.653 - 1.13i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.483 - 0.837i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.89 - 3.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.23T + 13T^{2} \)
17 \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.11 - 7.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.43 - 4.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.43T + 29T^{2} \)
31 \( 1 + (0.826 + 1.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.82 + 3.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.608T + 41T^{2} \)
43 \( 1 - 3.24T + 43T^{2} \)
47 \( 1 + (4.73 - 8.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.881 - 1.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.302 - 0.524i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.20 - 7.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.67 + 8.09i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.91T + 71T^{2} \)
73 \( 1 + (-5.29 - 9.17i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.52 - 9.56i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 + (7.97 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.28T + 97T^{2} \)
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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

−9.677495170609895139093219243072, −9.362520515378412500251932726534, −8.275300427350520323183966573954, −7.39977623781288338205605453089, −6.71079892397978200505280266450, −5.70102014671068801097605710590, −4.46685595676233433652976476661, −3.97056288149612698196030597493, −2.92595239889217734016759731734, −1.65320627189676001015885840659, 0.58563648681024461096056676082, 2.00956913326725844142495956015, 2.93697248952247832483339806475, 4.29119699910605774854643433348, 4.93327870262525385669620495812, 6.31534855272918987340424872967, 6.79319452086529202976077745601, 7.82151363353332852754014923422, 8.574161567062842069591218607230, 8.934678712329381285578436888591

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