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Properties

Label 2-4212-9.7-c1-0-10
Degree $2$
Conductor $4212$
Sign $-0.766 - 0.642i$
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  + (1.5 + 2.59i)5-s + (−1 + 1.73i)7-s + (−0.5 − 0.866i)13-s + 2·19-s + (3 + 5.19i)23-s + (−2 + 3.46i)25-s + (−3 + 5.19i)29-s + (−1 − 1.73i)31-s − 6·35-s + 2·37-s + (−3 − 5.19i)41-s + (−2.5 + 4.33i)43-s + (−1.5 + 2.59i)47-s + (1.50 + 2.59i)49-s + 12·53-s + ⋯
L(s)  = 1  + (0.670 + 1.16i)5-s + (−0.377 + 0.654i)7-s + (−0.138 − 0.240i)13-s + 0.458·19-s + (0.625 + 1.08i)23-s + (−0.400 + 0.692i)25-s + (−0.557 + 0.964i)29-s + (−0.179 − 0.311i)31-s − 1.01·35-s + 0.328·37-s + (−0.468 − 0.811i)41-s + (−0.381 + 0.660i)43-s + (−0.218 + 0.378i)47-s + (0.214 + 0.371i)49-s + 1.64·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4212\)    =    \(2^{2} \cdot 3^{4} \cdot 13\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(33.6329\)
Root analytic conductor: \(5.79939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4212} (2809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4212,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.585032861\)
\(L(\frac12)\) \(\approx\) \(1.585032861\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.872535777949813904920327214130, −7.80306209435411332816200460473, −7.12051726693156798969875470108, −6.55319484824430984045207521215, −5.65979179229739301857733738106, −5.32091317068583060352844825671, −3.96586651996121974444695453154, −3.05123190008121537158777032465, −2.56344593675158779120347666035, −1.44528350367981814555103324344, 0.44513288751530544199001631480, 1.42509888815941859900476682247, 2.45440802222864074514561007863, 3.58866027273235962139307343506, 4.44591835648572088719290421050, 5.10316482022856085825432345039, 5.82883311736297477491635772547, 6.67848050684711721708263004223, 7.32577463489354569079422802062, 8.291385189989464539296527322480

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