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Properties

Label 2-336-48.35-c1-0-18
Degree $2$
Conductor $336$
Sign $0.919 + 0.392i$
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  + (−1.30 − 0.533i)2-s + (1.26 − 1.18i)3-s + (1.42 + 1.39i)4-s + (1.97 + 1.97i)5-s + (−2.28 + 0.874i)6-s + 7-s + (−1.12 − 2.59i)8-s + (0.197 − 2.99i)9-s + (−1.53 − 3.64i)10-s + (−0.735 + 0.735i)11-s + (3.46 + 0.0758i)12-s + (2.83 + 2.83i)13-s + (−1.30 − 0.533i)14-s + (4.84 + 0.159i)15-s + (0.0887 + 3.99i)16-s + 3.89i·17-s + ⋯
L(s)  = 1  + (−0.925 − 0.377i)2-s + (0.730 − 0.683i)3-s + (0.714 + 0.699i)4-s + (0.885 + 0.885i)5-s + (−0.934 + 0.357i)6-s + 0.377·7-s + (−0.397 − 0.917i)8-s + (0.0659 − 0.997i)9-s + (−0.485 − 1.15i)10-s + (−0.221 + 0.221i)11-s + (0.999 + 0.0219i)12-s + (0.787 + 0.787i)13-s + (−0.349 − 0.142i)14-s + (1.25 + 0.0413i)15-s + (0.0221 + 0.999i)16-s + 0.944i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.919 + 0.392i$
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: \((2,\ 336,\ (\ :1/2),\ 0.919 + 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30273 - 0.266466i\)
\(L(\frac12)\) \(\approx\) \(1.30273 - 0.266466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.533i)T \)
3 \( 1 + (-1.26 + 1.18i)T \)
7 \( 1 - T \)
good5 \( 1 + (-1.97 - 1.97i)T + 5iT^{2} \)
11 \( 1 + (0.735 - 0.735i)T - 11iT^{2} \)
13 \( 1 + (-2.83 - 2.83i)T + 13iT^{2} \)
17 \( 1 - 3.89iT - 17T^{2} \)
19 \( 1 + (2.65 - 2.65i)T - 19iT^{2} \)
23 \( 1 + 4.59iT - 23T^{2} \)
29 \( 1 + (-6.93 + 6.93i)T - 29iT^{2} \)
31 \( 1 + 6.03iT - 31T^{2} \)
37 \( 1 + (7.23 - 7.23i)T - 37iT^{2} \)
41 \( 1 + 0.424T + 41T^{2} \)
43 \( 1 + (1.72 + 1.72i)T + 43iT^{2} \)
47 \( 1 + 3.45T + 47T^{2} \)
53 \( 1 + (7.07 + 7.07i)T + 53iT^{2} \)
59 \( 1 + (-1.95 + 1.95i)T - 59iT^{2} \)
61 \( 1 + (0.0653 + 0.0653i)T + 61iT^{2} \)
67 \( 1 + (-9.77 + 9.77i)T - 67iT^{2} \)
71 \( 1 - 8.50iT - 71T^{2} \)
73 \( 1 - 5.13iT - 73T^{2} \)
79 \( 1 + 2.65iT - 79T^{2} \)
83 \( 1 + (-0.666 - 0.666i)T + 83iT^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−11.35180635745488665618471852721, −10.34781151739458263184885823930, −9.744391046704769725122593312165, −8.513368353591046602980917667597, −8.059280059394168002517283316334, −6.59728866182386099556570400280, −6.36675141363229029904714594877, −3.89743965598589562894668479907, −2.52593840109617306428855152541, −1.69930308164609026226907208537, 1.48989228426485774001148453972, 2.98524055067690176500765403149, 4.93971083087816082488005442809, 5.56517741910607145409976745477, 7.05695063983623855134155960306, 8.284955594054542949234197036991, 8.804331847303168547721931339815, 9.519381898827890401493743554944, 10.46082177502638927082332250583, 11.11271888691258425293453050724

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