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Properties

Label 2-12-12.11-c5-0-6
Degree $2$
Conductor $12$
Sign $0.757 + 0.653i$
Analytic cond. $1.92460$
Root an. cond. $1.38730$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.41 − 1.63i)2-s + (−4.17 − 15.0i)3-s + (26.6 − 17.7i)4-s + 73.1i·5-s + (−47.2 − 74.4i)6-s + 51.8i·7-s + (115. − 139. i)8-s + (−208. + 125. i)9-s + (119. + 395. i)10-s − 371.·11-s + (−377. − 325. i)12-s + 424.·13-s + (84.8 + 280. i)14-s + (1.09e3 − 305. i)15-s + (394. − 944. i)16-s − 1.40e3i·17-s + ⋯
L(s)  = 1  + (0.957 − 0.289i)2-s + (−0.268 − 0.963i)3-s + (0.832 − 0.554i)4-s + 1.30i·5-s + (−0.535 − 0.844i)6-s + 0.399i·7-s + (0.636 − 0.771i)8-s + (−0.856 + 0.516i)9-s + (0.378 + 1.25i)10-s − 0.925·11-s + (−0.757 − 0.653i)12-s + 0.696·13-s + (0.115 + 0.382i)14-s + (1.26 − 0.350i)15-s + (0.385 − 0.922i)16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.757 + 0.653i$
Analytic conductor: \(1.92460\)
Root analytic conductor: \(1.38730\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :5/2),\ 0.757 + 0.653i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.69973 - 0.631734i\)
\(L(\frac12)\) \(\approx\) \(1.69973 - 0.631734i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.41 + 1.63i)T \)
3 \( 1 + (4.17 + 15.0i)T \)
good5 \( 1 - 73.1iT - 3.12e3T^{2} \)
7 \( 1 - 51.8iT - 1.68e4T^{2} \)
11 \( 1 + 371.T + 1.61e5T^{2} \)
13 \( 1 - 424.T + 3.71e5T^{2} \)
17 \( 1 + 1.40e3iT - 1.41e6T^{2} \)
19 \( 1 - 319. iT - 2.47e6T^{2} \)
23 \( 1 + 2.42e3T + 6.43e6T^{2} \)
29 \( 1 - 3.39e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.65e3iT - 2.86e7T^{2} \)
37 \( 1 - 395.T + 6.93e7T^{2} \)
41 \( 1 + 5.78e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.63e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.36e4T + 2.29e8T^{2} \)
53 \( 1 - 1.96e3iT - 4.18e8T^{2} \)
59 \( 1 - 7.86e3T + 7.14e8T^{2} \)
61 \( 1 + 9.36e3T + 8.44e8T^{2} \)
67 \( 1 - 3.78e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.03e4T + 1.80e9T^{2} \)
73 \( 1 + 6.20e4T + 2.07e9T^{2} \)
79 \( 1 + 6.42e4iT - 3.07e9T^{2} \)
83 \( 1 + 1.00e4T + 3.93e9T^{2} \)
89 \( 1 + 5.66e4iT - 5.58e9T^{2} \)
97 \( 1 + 2.23e4T + 8.58e9T^{2} \)
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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

−18.84722251429760650970022831858, −18.21961105913766412740928671099, −15.92407963208488900584120809931, −14.43590413777349031243221058349, −13.34724165040993531207209529216, −11.82211607925777921106010389274, −10.66114429824440499821982945807, −7.34573815542678746260068059937, −5.88452464376975547628511978710, −2.65981930108040445813116710220, 4.17191068651267564912068914668, 5.63974679889459828618839435776, 8.413237574211803470486473337022, 10.60175508668300287729382560623, 12.29181982813391927506832982143, 13.62892591175761809115767431093, 15.39801353926931646487715368465, 16.29539780185686006524345944757, 17.30475884128606982262207327023, 20.09167229157136759887217717353

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