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Properties

Label 2-336-7.4-c1-0-2
Degree $2$
Conductor $336$
Sign $0.991 - 0.126i$
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  + (0.5 − 0.866i)3-s + (1 + 1.73i)5-s + (2.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 13-s + 1.99·15-s + (0.5 + 0.866i)19-s + (2 − 1.73i)21-s + (0.500 − 0.866i)25-s − 0.999·27-s + 4·29-s + (4.5 − 7.79i)31-s + (0.999 + 1.73i)33-s + (1.00 + 5.19i)35-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.447 + 0.774i)5-s + (0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 0.277·13-s + 0.516·15-s + (0.114 + 0.198i)19-s + (0.436 − 0.377i)21-s + (0.100 − 0.173i)25-s − 0.192·27-s + 0.742·29-s + (0.808 − 1.39i)31-s + (0.174 + 0.301i)33-s + (0.169 + 0.878i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65947 + 0.105310i\)
\(L(\frac12)\) \(\approx\) \(1.65947 + 0.105310i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (8 + 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−11.61939431803824956055912989940, −10.64833204944929744237983947565, −9.807703268306811534641644660209, −8.596179846730826081126212669275, −7.80582835615406115758091385680, −6.80391003450114410908091692358, −5.82661766187358698849374095707, −4.56022737392867558812610682089, −2.91328961523927284058033257501, −1.79967279993630797846841592914, 1.47431717086110302589789240265, 3.22664450895118351570326821253, 4.69813009178533278507597604460, 5.26448461159680183834924613663, 6.69096001391946685916152534534, 8.163797806184969046578832758502, 8.575184208198096266151663157800, 9.683132609076294109829606803332, 10.58122589564776834266931025629, 11.40600450184990641011993417741

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