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Properties

Label 2-336-112.109-c1-0-7
Degree $2$
Conductor $336$
Sign $0.417 + 0.908i$
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  + (−1.02 − 0.978i)2-s + (−0.965 + 0.258i)3-s + (0.0862 + 1.99i)4-s + (−3.48 − 0.933i)5-s + (1.23 + 0.680i)6-s + (1.89 + 1.84i)7-s + (1.86 − 2.12i)8-s + (0.866 − 0.499i)9-s + (2.64 + 4.35i)10-s + (0.221 + 0.828i)11-s + (−0.600 − 1.90i)12-s + (0.343 + 0.343i)13-s + (−0.135 − 3.73i)14-s + 3.60·15-s + (−3.98 + 0.344i)16-s + (1.72 − 2.98i)17-s + ⋯
L(s)  = 1  + (−0.722 − 0.691i)2-s + (−0.557 + 0.149i)3-s + (0.0431 + 0.999i)4-s + (−1.55 − 0.417i)5-s + (0.506 + 0.277i)6-s + (0.717 + 0.696i)7-s + (0.659 − 0.751i)8-s + (0.288 − 0.166i)9-s + (0.836 + 1.37i)10-s + (0.0669 + 0.249i)11-s + (−0.173 − 0.550i)12-s + (0.0952 + 0.0952i)13-s + (−0.0362 − 0.999i)14-s + 0.930·15-s + (−0.996 + 0.0862i)16-s + (0.418 − 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.417 + 0.908i$
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: \((2,\ 336,\ (\ :1/2),\ 0.417 + 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506128 - 0.324598i\)
\(L(\frac12)\) \(\approx\) \(0.506128 - 0.324598i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.02 + 0.978i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-1.89 - 1.84i)T \)
good5 \( 1 + (3.48 + 0.933i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.221 - 0.828i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.343 - 0.343i)T + 13iT^{2} \)
17 \( 1 + (-1.72 + 2.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.07 + 7.72i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.56 + 1.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.67 - 4.67i)T + 29iT^{2} \)
31 \( 1 + (-3.93 + 6.82i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.23 + 0.867i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.63iT - 41T^{2} \)
43 \( 1 + (-4.39 + 4.39i)T - 43iT^{2} \)
47 \( 1 + (-4.92 - 8.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.936 - 3.49i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.16 + 11.8i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.30 - 8.60i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.05 + 0.283i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.97iT - 71T^{2} \)
73 \( 1 + (-3.36 - 1.94i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.17 - 2.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.9 - 11.9i)T + 83iT^{2} \)
89 \( 1 + (3.05 - 1.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−11.31074680336743800611434201826, −10.87697044712600480599787558816, −9.388591219049850990936475351893, −8.709101542841614051051547350304, −7.74775463810041461683759778165, −6.97975249569750766424813018439, −5.03472388810513625259525729692, −4.27408404341138728683272685657, −2.82338231885124247481820929161, −0.75984384463915104753267292608, 1.12525769564975687909208028449, 3.70837504209091015112898165637, 4.84517062862227913437535765572, 6.12942516225718134911197858430, 7.19263166691271865553584624166, 7.896090824720084097640167957601, 8.431148164782709162472744541322, 10.14007005843721210872964561592, 10.68693348330259207124785212500, 11.57132515450976692053149396671

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