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Properties

Label 2-4212-9.4-c1-0-13
Degree $2$
Conductor $4212$
Sign $-0.642 - 0.766i$
Analytic cond. $33.6329$
Root an. cond. $5.79939$
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  + (−1.09 + 1.89i)5-s + (1.39 + 2.41i)7-s + (1.09 + 1.89i)11-s + (0.5 − 0.866i)13-s − 1.73·17-s + 4.58·19-s + (−1.96 + 3.39i)23-s + (0.104 + 0.180i)25-s + (3.74 + 6.47i)29-s + (−0.895 + 1.55i)31-s − 6.10·35-s + 0.791·37-s + (1.55 − 2.68i)41-s + (−1.29 − 2.23i)43-s + (0.409 + 0.708i)47-s + ⋯
L(s)  = 1  + (−0.489 + 0.847i)5-s + (0.527 + 0.913i)7-s + (0.329 + 0.571i)11-s + (0.138 − 0.240i)13-s − 0.420·17-s + 1.05·19-s + (−0.408 + 0.708i)23-s + (0.0208 + 0.0361i)25-s + (0.694 + 1.20i)29-s + (−0.160 + 0.278i)31-s − 1.03·35-s + 0.130·37-s + (0.242 − 0.419i)41-s + (−0.196 − 0.341i)43-s + (0.0596 + 0.103i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4212\)    =    \(2^{2} \cdot 3^{4} \cdot 13\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(33.6329\)
Root analytic conductor: \(5.79939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4212} (1405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4212,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.658171188\)
\(L(\frac12)\) \(\approx\) \(1.658171188\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (1.09 - 1.89i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.39 - 2.41i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.09 - 1.89i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 - 4.58T + 19T^{2} \)
23 \( 1 + (1.96 - 3.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.74 - 6.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.895 - 1.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.791T + 37T^{2} \)
41 \( 1 + (-1.55 + 2.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.29 + 2.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.409 - 0.708i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.20T + 53T^{2} \)
59 \( 1 + (2.14 - 3.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.18 - 8.98i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.68 + 11.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.02T + 71T^{2} \)
73 \( 1 - 0.582T + 73T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.504 + 0.873i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + (-5.79 - 10.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.701777949407006778982191737180, −7.84177635686744977673128611264, −7.21392374193171318798092412488, −6.63024951381278678657078707354, −5.60191035874175733240824707147, −5.07839859609778301997527312366, −4.02086160390143878820446272511, −3.22038012264949443392892529015, −2.41026903944315915936544695294, −1.35747260897659921273548445890, 0.52481239350412921613529921101, 1.28363240895084219843119465283, 2.58822359194294230047400333700, 3.79161737491239992968883360596, 4.31704834106736036874289556473, 4.96830835430926632495155200556, 5.93270752111195978942638232825, 6.72819771236057758130992708881, 7.54726158902060935545256297955, 8.208487581671819736708141127325

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