Properties

Label 2-11-11.4-c3-0-1
Degree $2$
Conductor $11$
Sign $-0.150 + 0.988i$
Analytic cond. $0.649021$
Root an. cond. $0.805618$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.17 − 3.03i)2-s + (1.40 − 4.31i)3-s + (5.75 + 17.7i)4-s + (6.98 − 5.07i)5-s + (−18.9 + 13.7i)6-s + (0.513 + 1.58i)7-s + (16.9 − 52.1i)8-s + (5.15 + 3.74i)9-s − 44.5·10-s + (−26.2 + 25.3i)11-s + 84.5·12-s + (23.1 + 16.8i)13-s + (2.65 − 8.15i)14-s + (−12.1 − 37.2i)15-s + (−108. + 78.6i)16-s + (−6.36 + 4.62i)17-s + ⋯
L(s)  = 1  + (−1.47 − 1.07i)2-s + (0.270 − 0.831i)3-s + (0.719 + 2.21i)4-s + (0.624 − 0.453i)5-s + (−1.28 + 0.937i)6-s + (0.0277 + 0.0853i)7-s + (0.748 − 2.30i)8-s + (0.191 + 0.138i)9-s − 1.40·10-s + (−0.719 + 0.694i)11-s + 2.03·12-s + (0.494 + 0.358i)13-s + (0.0506 − 0.155i)14-s + (−0.208 − 0.642i)15-s + (−1.69 + 1.22i)16-s + (−0.0907 + 0.0659i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.150 + 0.988i$
Analytic conductor: \(0.649021\)
Root analytic conductor: \(0.805618\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :3/2),\ -0.150 + 0.988i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.365704 - 0.425396i\)
\(L(\frac12)\) \(\approx\) \(0.365704 - 0.425396i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (26.2 - 25.3i)T \)
good2 \( 1 + (4.17 + 3.03i)T + (2.47 + 7.60i)T^{2} \)
3 \( 1 + (-1.40 + 4.31i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (-6.98 + 5.07i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (-0.513 - 1.58i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (-23.1 - 16.8i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (6.36 - 4.62i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-14.8 + 45.6i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + 153.T + 1.21e4T^{2} \)
29 \( 1 + (-74.9 - 230. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (134. + 97.5i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (21.2 + 65.4i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-95.0 + 292. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 + 52.9T + 7.95e4T^{2} \)
47 \( 1 + (16.8 - 51.8i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (202. + 147. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (-117. - 360. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (565. - 410. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + 278.T + 3.00e5T^{2} \)
71 \( 1 + (-125. + 90.9i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (293. + 904. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (1.79 + 1.30i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-614. + 446. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 - 1.15e3T + 7.04e5T^{2} \)
97 \( 1 + (598. + 434. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−19.64656545974041844833800549140, −18.38417398666348919051138129766, −17.76671785230457100154887658197, −16.23137863381728838204098773632, −13.36083439366785730285016658716, −12.24860265273379933892510847996, −10.45476061694864452690370398450, −9.002443681269460045016356620077, −7.49525610904869238059665527215, −1.90141468123905850216987321031, 6.08933254733679359825471308870, 8.118902827648544348987814476459, 9.701436612108996873812731909927, 10.57941091108065317410326323355, 14.09504751333778429045235428592, 15.48333753302076647203353175001, 16.29794820778511096742650159002, 17.82169967982785171907834871499, 18.64767109659193475910909460395, 20.27590325450522842267171376876

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