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Properties

Label 2-2156-77.10-c1-0-30
Degree $2$
Conductor $2156$
Sign $-0.779 + 0.626i$
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  + (−1.93 − 1.11i)3-s + (1.93 − 1.11i)5-s + (1 + 1.73i)9-s + (−2.23 − 2.44i)11-s + 2.82·13-s − 5.00·15-s + (2.82 − 4.89i)17-s + (0.707 + 1.22i)19-s + (2.5 + 4.33i)23-s + 2.23i·27-s − 9.48i·29-s + (−1.93 − 1.11i)31-s + (1.59 + 7.24i)33-s + (−5.5 − 9.52i)37-s + (−5.47 − 3.16i)39-s + ⋯
L(s)  = 1  + (−1.11 − 0.645i)3-s + (0.866 − 0.499i)5-s + (0.333 + 0.577i)9-s + (−0.674 − 0.737i)11-s + 0.784·13-s − 1.29·15-s + (0.685 − 1.18i)17-s + (0.162 + 0.280i)19-s + (0.521 + 0.902i)23-s + 0.430i·27-s − 1.76i·29-s + (−0.347 − 0.200i)31-s + (0.278 + 1.26i)33-s + (−0.904 − 1.56i)37-s + (−0.877 − 0.506i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $-0.779 + 0.626i$
Analytic conductor: \(17.2157\)
Root analytic conductor: \(4.14918\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (2089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :1/2),\ -0.779 + 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.136053613\)
\(L(\frac12)\) \(\approx\) \(1.136053613\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 2.44i)T \)
good3 \( 1 + (1.93 + 1.11i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.93 + 1.11i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + (-2.82 + 4.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.707 - 1.22i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.5 - 4.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.48iT - 29T^{2} \)
31 \( 1 + (1.93 + 1.11i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 + 3.16iT - 43T^{2} \)
47 \( 1 + (-3.87 + 2.23i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4 - 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.93 - 1.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.12 - 3.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 + (-2.12 + 3.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.47 + 3.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.41T + 83T^{2} \)
89 \( 1 + (5.80 - 3.35i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.6iT - 97T^{2} \)
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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

−8.934611830262986225908610142721, −7.76825766476620038819920909621, −7.22970381042884071637210679395, −6.09141049273227445103324014045, −5.63478790867916277212639201665, −5.28167944102903505226994321627, −3.90312392049314877867174069886, −2.66922223013804542007264202401, −1.41327456226715787923580608623, −0.50339204821746332976866034296, 1.37689407416965410158882308237, 2.62397752958600527332693620323, 3.74067330543737308322947648016, 4.85378311295398470103251637984, 5.35088793977102980942212905683, 6.23484391997801072322910997782, 6.67150092265620839725903265297, 7.81311506956151805805050163496, 8.707912403186461430914370943849, 9.662800336730476257000192223803

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