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Properties

Label 2-336-48.35-c1-0-26
Degree $2$
Conductor $336$
Sign $0.915 - 0.401i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
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  + (1.29 + 0.576i)2-s + (−0.223 − 1.71i)3-s + (1.33 + 1.48i)4-s + (2.27 + 2.27i)5-s + (0.702 − 2.34i)6-s + 7-s + (0.866 + 2.69i)8-s + (−2.90 + 0.766i)9-s + (1.62 + 4.24i)10-s + (−1.53 + 1.53i)11-s + (2.25 − 2.62i)12-s + (−3.63 − 3.63i)13-s + (1.29 + 0.576i)14-s + (3.39 − 4.40i)15-s + (−0.433 + 3.97i)16-s + 1.81i·17-s + ⋯
L(s)  = 1  + (0.913 + 0.407i)2-s + (−0.128 − 0.991i)3-s + (0.667 + 0.744i)4-s + (1.01 + 1.01i)5-s + (0.286 − 0.958i)6-s + 0.377·7-s + (0.306 + 0.951i)8-s + (−0.966 + 0.255i)9-s + (0.513 + 1.34i)10-s + (−0.463 + 0.463i)11-s + (0.652 − 0.757i)12-s + (−1.00 − 1.00i)13-s + (0.345 + 0.154i)14-s + (0.876 − 1.13i)15-s + (−0.108 + 0.994i)16-s + 0.441i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.915 - 0.401i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.915 - 0.401i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33798 + 0.490057i\)
\(L(\frac12)\) \(\approx\) \(2.33798 + 0.490057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.576i)T \)
3 \( 1 + (0.223 + 1.71i)T \)
7 \( 1 - T \)
good5 \( 1 + (-2.27 - 2.27i)T + 5iT^{2} \)
11 \( 1 + (1.53 - 1.53i)T - 11iT^{2} \)
13 \( 1 + (3.63 + 3.63i)T + 13iT^{2} \)
17 \( 1 - 1.81iT - 17T^{2} \)
19 \( 1 + (-4.98 + 4.98i)T - 19iT^{2} \)
23 \( 1 + 7.21iT - 23T^{2} \)
29 \( 1 + (2.77 - 2.77i)T - 29iT^{2} \)
31 \( 1 + 4.14iT - 31T^{2} \)
37 \( 1 + (1.77 - 1.77i)T - 37iT^{2} \)
41 \( 1 - 0.517T + 41T^{2} \)
43 \( 1 + (4.67 + 4.67i)T + 43iT^{2} \)
47 \( 1 - 1.19T + 47T^{2} \)
53 \( 1 + (7.94 + 7.94i)T + 53iT^{2} \)
59 \( 1 + (-1.94 + 1.94i)T - 59iT^{2} \)
61 \( 1 + (-2.94 - 2.94i)T + 61iT^{2} \)
67 \( 1 + (7.85 - 7.85i)T - 67iT^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 - 15.7iT - 73T^{2} \)
79 \( 1 - 5.40iT - 79T^{2} \)
83 \( 1 + (7.12 + 7.12i)T + 83iT^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−11.80591122161400500491601100872, −10.90256067126623124875844001571, −10.01984695350522100231235650304, −8.400216566903699439171104665692, −7.34578236541985811547840244562, −6.84397232442729278228619174061, −5.75088018727081248620082708573, −5.00892514000255710961369979161, −2.92901643270490743143126829740, −2.23157351098485906948784561898, 1.73631864428956944852093189942, 3.30279805241393173328034823616, 4.67774945633580543970979788369, 5.26815944727494706405997988448, 6.01077830408885229030650236541, 7.66635936459178851424858864319, 9.301424145056651205234041815182, 9.591060561884380763622750010999, 10.60027373732769359976066935383, 11.67875944990076703809979232417

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