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Properties

Label 2-336-84.23-c1-0-12
Degree $2$
Conductor $336$
Sign $-0.922 + 0.386i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (−2.5 + 0.866i)7-s + (1.5 − 2.59i)9-s − 5·13-s + (−7.5 − 4.33i)19-s + (3 − 3.46i)21-s + (−2.5 − 4.33i)25-s + 5.19i·27-s + (1.5 − 0.866i)31-s + (−5.5 + 9.52i)37-s + (7.5 − 4.33i)39-s + 1.73i·43-s + (5.5 − 4.33i)49-s + 15·57-s + (−7 + 12.1i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (−0.944 + 0.327i)7-s + (0.5 − 0.866i)9-s − 1.38·13-s + (−1.72 − 0.993i)19-s + (0.654 − 0.755i)21-s + (−0.5 − 0.866i)25-s + 0.999i·27-s + (0.269 − 0.155i)31-s + (−0.904 + 1.56i)37-s + (1.20 − 0.693i)39-s + 0.264i·43-s + (0.785 − 0.618i)49-s + 1.98·57-s + (−0.896 + 1.55i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.922 + 0.386i$
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: \(1\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.922 + 0.386i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.5 + 4.33i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.5 + 7.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-8.5 - 14.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 2.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−11.11180629623715868657116193614, −10.14479362364973366688747831264, −9.592427830114667685884079120474, −8.488810895574837430462768106127, −6.94499583564325460533144137913, −6.32539805574293986961846929311, −5.13040568508984478534417546846, −4.16518528755681873745492429103, −2.61890478527666292285579917269, 0, 2.13156604298823273728923710699, 3.89446156586370896030199234183, 5.18070307832391314090556342001, 6.24145103685537273390467546715, 7.03859651262187464541424817721, 7.940600013302855285663268770698, 9.370783447965846290897680573807, 10.26930492988149522836326983773, 10.95246137292944242835692521418

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