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Properties

Label 2-336-112.109-c1-0-22
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  + (1.37 − 0.322i)2-s + (−0.965 + 0.258i)3-s + (1.79 − 0.887i)4-s + (2.43 + 0.653i)5-s + (−1.24 + 0.667i)6-s + (−1.71 − 2.01i)7-s + (2.18 − 1.79i)8-s + (0.866 − 0.499i)9-s + (3.56 + 0.114i)10-s + (−0.0306 − 0.114i)11-s + (−1.50 + 1.32i)12-s + (2.21 + 2.21i)13-s + (−3.00 − 2.22i)14-s − 2.52·15-s + (2.42 − 3.18i)16-s + (−2.45 + 4.26i)17-s + ⋯
L(s)  = 1  + (0.973 − 0.227i)2-s + (−0.557 + 0.149i)3-s + (0.896 − 0.443i)4-s + (1.09 + 0.292i)5-s + (−0.508 + 0.272i)6-s + (−0.647 − 0.762i)7-s + (0.771 − 0.635i)8-s + (0.288 − 0.166i)9-s + (1.12 + 0.0361i)10-s + (−0.00925 − 0.0345i)11-s + (−0.433 + 0.381i)12-s + (0.613 + 0.613i)13-s + (−0.803 − 0.594i)14-s − 0.651·15-s + (0.606 − 0.795i)16-s + (−0.596 + 1.03i)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: $\chi_{336} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.919 + 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20419 - 0.451211i\)
\(L(\frac12)\) \(\approx\) \(2.20419 - 0.451211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.322i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (1.71 + 2.01i)T \)
good5 \( 1 + (-2.43 - 0.653i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.0306 + 0.114i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.21 - 2.21i)T + 13iT^{2} \)
17 \( 1 + (2.45 - 4.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.896 + 3.34i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.22 + 1.28i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.376 - 0.376i)T + 29iT^{2} \)
31 \( 1 + (4.09 - 7.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.27 + 2.21i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.368iT - 41T^{2} \)
43 \( 1 + (3.90 - 3.90i)T - 43iT^{2} \)
47 \( 1 + (-0.566 - 0.981i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.15 + 11.7i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.157 + 0.586i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.57 - 13.3i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (6.83 - 1.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.37iT - 71T^{2} \)
73 \( 1 + (4.59 + 2.65i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.74 - 4.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.86 + 2.86i)T + 83iT^{2} \)
89 \( 1 + (-3.57 + 2.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−11.41168534076266388291047832983, −10.62546168755956748357048281562, −10.10753167160410032063761218339, −8.949222768802528490782499393021, −7.01119603751821630609678892459, −6.51641101245857909173653485119, −5.61347261612643515540706739423, −4.44039485346197966567133446764, −3.28525210054335121589270393948, −1.68322485599635058582313451672, 1.96843059884239340447036656601, 3.32376290705505447717165828917, 4.96602689331051829335731963968, 5.75023330809684720958492416441, 6.32591069040401622609047096227, 7.48227960104889701351198198318, 8.868101375535750941459224356907, 9.859514220658598707032204830557, 10.89728742730232553687963240020, 11.87230516294980549861476225263

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