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Properties

Label 2-2156-7.2-c1-0-18
Degree $2$
Conductor $2156$
Sign $0.991 + 0.126i$
Analytic cond. $17.2157$
Root an. cond. $4.14918$
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  + (0.160 + 0.278i)3-s + (−1.94 + 3.37i)5-s + (1.44 − 2.50i)9-s + (−0.5 − 0.866i)11-s − 0.218·13-s − 1.25·15-s + (−2.43 − 4.21i)17-s + (0.321 − 0.557i)19-s + (3.26 − 5.66i)23-s + (−5.09 − 8.81i)25-s + 1.89·27-s + 10.4·29-s + (1.51 + 2.62i)31-s + (0.160 − 0.278i)33-s + (0.948 − 1.64i)37-s + ⋯
L(s)  = 1  + (0.0928 + 0.160i)3-s + (−0.871 + 1.50i)5-s + (0.482 − 0.836i)9-s + (−0.150 − 0.261i)11-s − 0.0605·13-s − 0.323·15-s + (−0.589 − 1.02i)17-s + (0.0737 − 0.127i)19-s + (0.681 − 1.18i)23-s + (−1.01 − 1.76i)25-s + 0.364·27-s + 1.93·29-s + (0.272 + 0.472i)31-s + (0.0279 − 0.0484i)33-s + (0.155 − 0.270i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(17.2157\)
Root analytic conductor: \(4.14918\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.426257493\)
\(L(\frac12)\) \(\approx\) \(1.426257493\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.160 - 0.278i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.94 - 3.37i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 0.218T + 13T^{2} \)
17 \( 1 + (2.43 + 4.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.321 + 0.557i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.26 + 5.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 + (-1.51 - 2.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.948 + 1.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.86T + 41T^{2} \)
43 \( 1 + 1.35T + 43T^{2} \)
47 \( 1 + (-2.78 + 4.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.89 - 6.74i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.83 - 6.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.00 + 6.93i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.26 + 2.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 + (-4.43 - 7.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.57 - 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.72T + 83T^{2} \)
89 \( 1 + (-5.30 + 9.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.746T + 97T^{2} \)
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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

−8.954230986069185474751728195966, −8.331584983545290254194983829200, −7.22058892981886070720661296939, −6.88058111322526611873778353957, −6.24170336968499550473430869006, −4.84454343867032426761033347930, −4.08187080413008127069325662175, −3.11066521037368758677767850212, −2.60807007762075833575688387093, −0.63376028209176066330298216705, 0.990553545475000579962355740144, 1.97881037082722025692354397981, 3.42603938634556892874002736490, 4.51440583412259033568327453378, 4.78185682215433250577209714816, 5.79584642693067227029280620183, 6.95554026952207755992985090767, 7.72963587311613982553051174851, 8.341282070037325762990030861303, 8.820801911501504162237685163469

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