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Properties

Label 2-336-112.109-c1-0-11
Degree $2$
Conductor $336$
Sign $0.177 - 0.984i$
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  + (1.07 + 0.923i)2-s + (0.965 − 0.258i)3-s + (0.296 + 1.97i)4-s + (−1.04 − 0.279i)5-s + (1.27 + 0.614i)6-s + (−0.0520 + 2.64i)7-s + (−1.50 + 2.39i)8-s + (0.866 − 0.499i)9-s + (−0.859 − 1.26i)10-s + (0.0302 + 0.112i)11-s + (0.797 + 1.83i)12-s + (3.16 + 3.16i)13-s + (−2.49 + 2.78i)14-s − 1.07·15-s + (−3.82 + 1.17i)16-s + (2.36 − 4.10i)17-s + ⋯
L(s)  = 1  + (0.757 + 0.652i)2-s + (0.557 − 0.149i)3-s + (0.148 + 0.988i)4-s + (−0.466 − 0.124i)5-s + (0.520 + 0.250i)6-s + (−0.0196 + 0.999i)7-s + (−0.533 + 0.845i)8-s + (0.288 − 0.166i)9-s + (−0.271 − 0.399i)10-s + (0.00911 + 0.0340i)11-s + (0.230 + 0.529i)12-s + (0.876 + 0.876i)13-s + (−0.667 + 0.744i)14-s − 0.278·15-s + (−0.956 + 0.292i)16-s + (0.574 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.177 - 0.984i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.177 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66564 + 1.39195i\)
\(L(\frac12)\) \(\approx\) \(1.66564 + 1.39195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 - 0.923i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (0.0520 - 2.64i)T \)
good5 \( 1 + (1.04 + 0.279i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.0302 - 0.112i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.16 - 3.16i)T + 13iT^{2} \)
17 \( 1 + (-2.36 + 4.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.640 + 2.38i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.25 + 1.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.88 + 1.88i)T + 29iT^{2} \)
31 \( 1 + (-2.82 + 4.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.30 + 0.349i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 12.7iT - 41T^{2} \)
43 \( 1 + (8.99 - 8.99i)T - 43iT^{2} \)
47 \( 1 + (-3.95 - 6.84i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.52 + 5.68i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.863 - 3.22i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.633 - 2.36i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-7.79 + 2.08i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.29iT - 71T^{2} \)
73 \( 1 + (-1.73 - 1.00i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.83 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.30 + 6.30i)T + 83iT^{2} \)
89 \( 1 + (-1.47 + 0.851i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 18.5T + 97T^{2} \)
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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

−11.85035940534451835869533820597, −11.35160638847872928021206026374, −9.517453515791961980411909911347, −8.738048702206769772828690379288, −7.933631058398998374875911876445, −6.92687189107328103397151961107, −5.91987898658764762516611571556, −4.75816771121397592226325090138, −3.62168591172605876055085927388, −2.41818305148937183060499138262, 1.39435910827385867435546293023, 3.35815982507571265565170726932, 3.76953012378366632153686449136, 5.12974425316267469696394038547, 6.37195263151376370177010322384, 7.56597007458574297762851314311, 8.517450435246759498606846230897, 9.906792188803844588521884897513, 10.49097196097247143702017158044, 11.28467058137426352435212800402

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