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Properties

Label 2-336-48.11-c1-0-35
Degree $2$
Conductor $336$
Sign $0.955 + 0.293i$
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  + (1.33 + 0.456i)2-s + (0.277 − 1.70i)3-s + (1.58 + 1.22i)4-s + (0.459 − 0.459i)5-s + (1.15 − 2.16i)6-s + 7-s + (1.55 + 2.35i)8-s + (−2.84 − 0.947i)9-s + (0.824 − 0.405i)10-s + (1.43 + 1.43i)11-s + (2.52 − 2.36i)12-s + (−1.18 + 1.18i)13-s + (1.33 + 0.456i)14-s + (−0.658 − 0.912i)15-s + (1.00 + 3.87i)16-s − 7.20i·17-s + ⋯
L(s)  = 1  + (0.946 + 0.323i)2-s + (0.159 − 0.987i)3-s + (0.791 + 0.611i)4-s + (0.205 − 0.205i)5-s + (0.470 − 0.882i)6-s + 0.377·7-s + (0.551 + 0.834i)8-s + (−0.948 − 0.315i)9-s + (0.260 − 0.128i)10-s + (0.432 + 0.432i)11-s + (0.730 − 0.683i)12-s + (−0.328 + 0.328i)13-s + (0.357 + 0.122i)14-s + (−0.169 − 0.235i)15-s + (0.252 + 0.967i)16-s − 1.74i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.955 + 0.293i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.955 + 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44504 - 0.367489i\)
\(L(\frac12)\) \(\approx\) \(2.44504 - 0.367489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 - 0.456i)T \)
3 \( 1 + (-0.277 + 1.70i)T \)
7 \( 1 - T \)
good5 \( 1 + (-0.459 + 0.459i)T - 5iT^{2} \)
11 \( 1 + (-1.43 - 1.43i)T + 11iT^{2} \)
13 \( 1 + (1.18 - 1.18i)T - 13iT^{2} \)
17 \( 1 + 7.20iT - 17T^{2} \)
19 \( 1 + (2.23 + 2.23i)T + 19iT^{2} \)
23 \( 1 - 0.540iT - 23T^{2} \)
29 \( 1 + (4.14 + 4.14i)T + 29iT^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 + (1.36 + 1.36i)T + 37iT^{2} \)
41 \( 1 + 2.16T + 41T^{2} \)
43 \( 1 + (6.40 - 6.40i)T - 43iT^{2} \)
47 \( 1 - 7.63T + 47T^{2} \)
53 \( 1 + (-2.29 + 2.29i)T - 53iT^{2} \)
59 \( 1 + (6.32 + 6.32i)T + 59iT^{2} \)
61 \( 1 + (-0.637 + 0.637i)T - 61iT^{2} \)
67 \( 1 + (2.14 + 2.14i)T + 67iT^{2} \)
71 \( 1 - 14.6iT - 71T^{2} \)
73 \( 1 + 0.233iT - 73T^{2} \)
79 \( 1 + 4.93iT - 79T^{2} \)
83 \( 1 + (-1.76 + 1.76i)T - 83iT^{2} \)
89 \( 1 + 7.79T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−11.77208281563310427158927381974, −11.11329340836923425543468566691, −9.453710300796179136273052839328, −8.458814358478229065666413231866, −7.26265101680950464271536942260, −6.87659451792784087042696893684, −5.56875517401903747168381700307, −4.65413261149736243117246155314, −3.06193591179387705456018065747, −1.80875955373711770990565408104, 2.13362411532303450995665374643, 3.56893597074795543131773497464, 4.32895386464083205968113431960, 5.57948004259368864759465963374, 6.28340588590680697909576210210, 7.85868591610890110117260444023, 8.934820908367293493891404757049, 10.22246938012142263964111587380, 10.59193312393283358692378164392, 11.54660824548286886237270572844

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