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Properties

Label 2-4212-9.7-c1-0-28
Degree $2$
Conductor $4212$
Sign $0.939 - 0.342i$
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  + (2 + 3.46i)5-s + (1 − 1.73i)7-s + (2 − 3.46i)11-s + (−0.5 − 0.866i)13-s + 2·17-s − 2·19-s + (−5.49 + 9.52i)25-s + (3 − 5.19i)29-s + (5 + 8.66i)31-s + 7.99·35-s + 10·37-s + (−4 − 6.92i)41-s + (−2 + 3.46i)43-s + (2 − 3.46i)47-s + (1.50 + 2.59i)49-s + ⋯
L(s)  = 1  + (0.894 + 1.54i)5-s + (0.377 − 0.654i)7-s + (0.603 − 1.04i)11-s + (−0.138 − 0.240i)13-s + 0.485·17-s − 0.458·19-s + (−1.09 + 1.90i)25-s + (0.557 − 0.964i)29-s + (0.898 + 1.55i)31-s + 1.35·35-s + 1.64·37-s + (−0.624 − 1.08i)41-s + (−0.304 + 0.528i)43-s + (0.291 − 0.505i)47-s + (0.214 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4212\)    =    \(2^{2} \cdot 3^{4} \cdot 13\)
Sign: $0.939 - 0.342i$
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: \((2,\ 4212,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.620154302\)
\(L(\frac12)\) \(\approx\) \(2.620154302\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (4 + 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2 + 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.318722787635579657883370759815, −7.67760689707538305969090227413, −6.78851364991357310204536516151, −6.35999163851711778941089101648, −5.72381874997567182015928709245, −4.69407514231211404461411533710, −3.62878154429532031765140938907, −3.02486775436371686068199275395, −2.12619821714637538668494324312, −0.951642609230262267305226911386, 0.986201784551478799317341789883, 1.81413736804662595977172651594, 2.56371537692863679472990010296, 4.11854933526680175591099442388, 4.68283184720550380875812724769, 5.30923009211381278515036037747, 6.05129178862079530266475343784, 6.75387822901666802676019035979, 7.894394610814297621259935060266, 8.426307134069867592713667197275

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