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Properties

Label 2-1372-7.2-c1-0-8
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  + (0.325 + 0.562i)3-s + (−0.794 + 1.37i)5-s + (1.28 − 2.23i)9-s + (−0.336 − 0.582i)11-s − 0.938·13-s − 1.03·15-s + (3.21 + 5.57i)17-s + (1.19 − 2.06i)19-s + (0.805 − 1.39i)23-s + (1.23 + 2.14i)25-s + 3.62·27-s + 4.64·29-s + (4.23 + 7.33i)31-s + (0.218 − 0.378i)33-s + (−2.63 + 4.56i)37-s + ⋯
L(s)  = 1  + (0.187 + 0.325i)3-s + (−0.355 + 0.615i)5-s + (0.429 − 0.744i)9-s + (−0.101 − 0.175i)11-s − 0.260·13-s − 0.266·15-s + (0.780 + 1.35i)17-s + (0.273 − 0.474i)19-s + (0.168 − 0.291i)23-s + (0.247 + 0.429i)25-s + 0.697·27-s + 0.861·29-s + (0.760 + 1.31i)31-s + (0.0380 − 0.0659i)33-s + (−0.433 + 0.750i)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

\(L(1)\) \(\approx\) \(1.693763414\)
\(L(\frac12)\) \(\approx\) \(1.693763414\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.325 - 0.562i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.794 - 1.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.336 + 0.582i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.938T + 13T^{2} \)
17 \( 1 + (-3.21 - 5.57i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.19 + 2.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.805 + 1.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.64T + 29T^{2} \)
31 \( 1 + (-4.23 - 7.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.63 - 4.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.14T + 41T^{2} \)
43 \( 1 - 0.355T + 43T^{2} \)
47 \( 1 + (-2.41 + 4.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.91 + 6.77i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.15 - 8.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.12 - 5.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.54 - 9.60i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.93T + 71T^{2} \)
73 \( 1 + (-6.60 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.799 - 1.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.74T + 83T^{2} \)
89 \( 1 + (6.06 - 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−10.02675485496248210100384436704, −8.776428714885695189904281483450, −8.291072191664967758901620916755, −7.09828763270761731693572724424, −6.66580223599942076266417505459, −5.55716093005555064727337734265, −4.50921904305464681274241664582, −3.55258070307952109984411064962, −2.88758593968247339693175041562, −1.23028201836778799861035353332, 0.801380379271367291025907003332, 2.13938990372742366501259128467, 3.23950636349053053586303516137, 4.56226672198997815914885402620, 5.03179401746093062274134547040, 6.16977787690394817381970142184, 7.31491240889393860309252417748, 7.74831212199593584108054404677, 8.513191702346805182367303166468, 9.503873910071722863523565660943

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