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Properties

Label 2-4212-9.7-c1-0-18
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: $\chi_{4212} (2809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4212,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.282709722\)
\(L(\frac12)\) \(\approx\) \(2.282709722\)
\(L(\frac{3}{2})\) 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

−8.395471075481528245474751034982, −7.49158768942231121782310058683, −7.22397149321775094754046035195, −6.43803214062265288463398918326, −5.46373652856855912143621191494, −4.86869246063494159444299438142, −3.81841601409144602043540666280, −3.11955875593102390120378055620, −2.05760550171621942599819072831, −1.11077252854863385083026579180, 0.73495432701089728908479641709, 1.77914198740528016413382152446, 2.71514407040210959892180682105, 3.65206865576187069743777057159, 4.84162912950071093393339214606, 5.34615091009178904211946392058, 5.83061597108848188451431680199, 6.76835419500932615569079919124, 7.895376618407963672757565768295, 8.229026351914022808895281706644

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