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Properties

Label 2-12-4.3-c4-0-0
Degree $2$
Conductor $12$
Sign $0.150 - 0.988i$
Analytic cond. $1.24043$
Root an. cond. $1.11375$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.302 + 3.98i)2-s + 5.19i·3-s + (−15.8 − 2.41i)4-s + 34.8·5-s + (−20.7 − 1.57i)6-s − 43.0i·7-s + (14.4 − 62.3i)8-s − 27·9-s + (−10.5 + 138. i)10-s + 65.2i·11-s + (12.5 − 82.1i)12-s − 99.0·13-s + (171. + 13.0i)14-s + 181. i·15-s + (244. + 76.4i)16-s − 207.·17-s + ⋯
L(s)  = 1  + (−0.0756 + 0.997i)2-s + 0.577i·3-s + (−0.988 − 0.150i)4-s + 1.39·5-s + (−0.575 − 0.0437i)6-s − 0.878i·7-s + (0.225 − 0.974i)8-s − 0.333·9-s + (−0.105 + 1.38i)10-s + 0.539i·11-s + (0.0871 − 0.570i)12-s − 0.586·13-s + (0.875 + 0.0664i)14-s + 0.804i·15-s + (0.954 + 0.298i)16-s − 0.718·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.150 - 0.988i$
Analytic conductor: \(1.24043\)
Root analytic conductor: \(1.11375\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :2),\ 0.150 - 0.988i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.835283 + 0.717415i\)
\(L(\frac12)\) \(\approx\) \(0.835283 + 0.717415i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.302 - 3.98i)T \)
3 \( 1 - 5.19iT \)
good5 \( 1 - 34.8T + 625T^{2} \)
7 \( 1 + 43.0iT - 2.40e3T^{2} \)
11 \( 1 - 65.2iT - 1.46e4T^{2} \)
13 \( 1 + 99.0T + 2.85e4T^{2} \)
17 \( 1 + 207.T + 8.35e4T^{2} \)
19 \( 1 + 569. iT - 1.30e5T^{2} \)
23 \( 1 - 371. iT - 2.79e5T^{2} \)
29 \( 1 - 423.T + 7.07e5T^{2} \)
31 \( 1 - 1.17e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.44e3T + 1.87e6T^{2} \)
41 \( 1 + 265.T + 2.82e6T^{2} \)
43 \( 1 - 699. iT - 3.41e6T^{2} \)
47 \( 1 - 1.25e3iT - 4.87e6T^{2} \)
53 \( 1 - 787.T + 7.89e6T^{2} \)
59 \( 1 + 3.01e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.51e3T + 1.38e7T^{2} \)
67 \( 1 + 5.21e3iT - 2.01e7T^{2} \)
71 \( 1 - 4.69e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.40e3T + 2.83e7T^{2} \)
79 \( 1 + 7.45e3iT - 3.89e7T^{2} \)
83 \( 1 - 8.95e3iT - 4.74e7T^{2} \)
89 \( 1 + 616.T + 6.27e7T^{2} \)
97 \( 1 - 1.37e4T + 8.85e7T^{2} \)
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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

−19.80409457020598747465178587368, −17.65561086366146165583328136514, −17.28972121736108648154944507251, −15.71325364347190545992028818705, −14.25169970326605343776020203273, −13.26630440922724312702570987254, −10.31970931103209328325156031806, −9.191546723912840722951423572117, −6.86371007993304063065754336602, −4.94391572785138053514189951270, 2.19838578255000481774688793477, 5.74892451087126920716738699033, 8.714537531290301440109738135882, 10.16515386371985889655554108940, 12.00448445958351114897360335675, 13.24138420205178385669881740226, 14.39756887227521205062159558897, 17.02558031384986871131911013558, 18.15794452648903551197687389496, 19.01412484068610789336533989452

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