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Properties

Label 2-12-12.11-c17-0-16
Degree $2$
Conductor $12$
Sign $0.877 + 0.479i$
Analytic cond. $21.9866$
Root an. cond. $4.68899$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (97.3 − 348. i)2-s + (1.13e4 − 502. i)3-s + (−1.12e5 − 6.78e4i)4-s + 1.05e6i·5-s + (9.29e5 − 4.00e6i)6-s − 1.99e5i·7-s + (−3.45e7 + 3.24e7i)8-s + (1.28e8 − 1.14e7i)9-s + (3.67e8 + 1.02e8i)10-s + 9.55e8·11-s + (−1.30e9 − 7.14e8i)12-s + 1.34e9·13-s + (−6.94e7 − 1.93e7i)14-s + (5.30e8 + 1.19e10i)15-s + (7.96e9 + 1.52e10i)16-s + 3.57e10i·17-s + ⋯
L(s)  = 1  + (0.268 − 0.963i)2-s + (0.999 − 0.0442i)3-s + (−0.855 − 0.517i)4-s + 1.20i·5-s + (0.225 − 0.974i)6-s − 0.0130i·7-s + (−0.728 + 0.684i)8-s + (0.996 − 0.0884i)9-s + (1.16 + 0.324i)10-s + 1.34·11-s + (−0.877 − 0.479i)12-s + 0.456·13-s + (−0.0125 − 0.00350i)14-s + (0.0534 + 1.20i)15-s + (0.463 + 0.885i)16-s + 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.877 + 0.479i$
Analytic conductor: \(21.9866\)
Root analytic conductor: \(4.68899\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :17/2),\ 0.877 + 0.479i)\)

Particular Values

\(L(9)\) \(\approx\) \(3.02662 - 0.772854i\)
\(L(\frac12)\) \(\approx\) \(3.02662 - 0.772854i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-97.3 + 348. i)T \)
3 \( 1 + (-1.13e4 + 502. i)T \)
good5 \( 1 - 1.05e6iT - 7.62e11T^{2} \)
7 \( 1 + 1.99e5iT - 2.32e14T^{2} \)
11 \( 1 - 9.55e8T + 5.05e17T^{2} \)
13 \( 1 - 1.34e9T + 8.65e18T^{2} \)
17 \( 1 - 3.57e10iT - 8.27e20T^{2} \)
19 \( 1 + 4.31e10iT - 5.48e21T^{2} \)
23 \( 1 - 4.34e11T + 1.41e23T^{2} \)
29 \( 1 + 4.71e12iT - 7.25e24T^{2} \)
31 \( 1 - 4.05e12iT - 2.25e25T^{2} \)
37 \( 1 + 2.04e13T + 4.56e26T^{2} \)
41 \( 1 - 1.64e13iT - 2.61e27T^{2} \)
43 \( 1 - 8.92e13iT - 5.87e27T^{2} \)
47 \( 1 + 2.84e14T + 2.66e28T^{2} \)
53 \( 1 + 2.90e14iT - 2.05e29T^{2} \)
59 \( 1 - 1.13e15T + 1.27e30T^{2} \)
61 \( 1 - 2.13e14T + 2.24e30T^{2} \)
67 \( 1 - 6.31e15iT - 1.10e31T^{2} \)
71 \( 1 - 2.32e15T + 2.96e31T^{2} \)
73 \( 1 + 6.61e15T + 4.74e31T^{2} \)
79 \( 1 + 9.86e15iT - 1.81e32T^{2} \)
83 \( 1 + 2.21e16T + 4.21e32T^{2} \)
89 \( 1 - 8.78e14iT - 1.37e33T^{2} \)
97 \( 1 - 1.38e16T + 5.95e33T^{2} \)
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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

−15.07388303375720469045613385207, −14.34847081211929721126347900018, −13.10659572535944356306443242749, −11.38313759684840875204655937626, −10.05908981380874951086323968746, −8.668413382663570560689562166260, −6.61428000556783744504056509838, −4.00606732162810114807828030091, −2.92362849398599047274248970720, −1.46249821494191179621519961707, 1.10846276743689782937239505854, 3.62404442373614364296360510885, 4.98581306099440786538593941284, 7.02250741708585400839032823071, 8.633447983558150878912845009043, 9.296413630928137923402204240355, 12.32131580648382395221962634939, 13.52656694362679052659605070465, 14.60992205663950467105559925482, 15.99720444692639117202360225229

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