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Properties

Label 2-336-84.23-c1-0-7
Degree $2$
Conductor $336$
Sign $0.506 + 0.862i$
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  + (−1.37 − 1.05i)3-s + (0.581 + 0.335i)5-s + (2.63 − 0.209i)7-s + (0.754 + 2.90i)9-s + (−2.62 − 4.54i)11-s + 2·13-s + (−0.440 − 1.07i)15-s + (3.64 − 2.10i)17-s + (1.13 + 0.656i)19-s + (−3.83 − 2.50i)21-s + (1.60 − 2.77i)23-s + (−2.27 − 3.94i)25-s + (2.04 − 4.77i)27-s − 6.22i·29-s + (1.5 − 0.866i)31-s + ⋯
L(s)  = 1  + (−0.791 − 0.611i)3-s + (0.259 + 0.150i)5-s + (0.996 − 0.0791i)7-s + (0.251 + 0.967i)9-s + (−0.791 − 1.37i)11-s + 0.554·13-s + (−0.113 − 0.277i)15-s + (0.884 − 0.510i)17-s + (0.260 + 0.150i)19-s + (−0.836 − 0.547i)21-s + (0.334 − 0.578i)23-s + (−0.454 − 0.788i)25-s + (0.393 − 0.919i)27-s − 1.15i·29-s + (0.269 − 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.506 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00443 - 0.574662i\)
\(L(\frac12)\) \(\approx\) \(1.00443 - 0.574662i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.37 + 1.05i)T \)
7 \( 1 + (-2.63 + 0.209i)T \)
good5 \( 1 + (-0.581 - 0.335i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.62 + 4.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-3.64 + 2.10i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.13 - 0.656i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.60 + 2.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.22iT - 29T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.13 + 5.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.76iT - 41T^{2} \)
43 \( 1 - 9.55iT - 43T^{2} \)
47 \( 1 + (4.80 - 8.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.1 - 5.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.78 - 6.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.86 + 3.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.58 - 1.49i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.08T + 71T^{2} \)
73 \( 1 + (0.137 + 0.238i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 2.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.04T + 83T^{2} \)
89 \( 1 + (-12.1 - 6.98i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.27T + 97T^{2} \)
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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

−11.23784189776351263601368617574, −10.87890348363429748550052821492, −9.716571458305823533318884411711, −8.094879477682084151195592105010, −7.87506212937160498254716298643, −6.31248632135927759289177346345, −5.66512147156590385490568534234, −4.57461653949000703378427235498, −2.72702075575273472176346051101, −1.03317023366546928274615940393, 1.64785015284062591901682644105, 3.67244289494838015035731114312, 5.03081132733654247450566037113, 5.40550409584819355863691980448, 6.87495484302867490131849646341, 7.892443761147928115349144414267, 9.083194745294306943726540377389, 10.05878576879182194633689103185, 10.70862502917557108108115487107, 11.65940220539761174888501662637

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