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Properties

Label 2-336-112.109-c1-0-25
Degree $2$
Conductor $336$
Sign $0.931 + 0.362i$
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  + (1.40 − 0.142i)2-s + (0.965 − 0.258i)3-s + (1.95 − 0.400i)4-s + (−2.36 − 0.634i)5-s + (1.32 − 0.501i)6-s + (2.51 + 0.828i)7-s + (2.70 − 0.842i)8-s + (0.866 − 0.499i)9-s + (−3.42 − 0.555i)10-s + (0.753 + 2.81i)11-s + (1.78 − 0.894i)12-s + (−2.60 − 2.60i)13-s + (3.65 + 0.808i)14-s − 2.45·15-s + (3.67 − 1.56i)16-s + (−1.53 + 2.65i)17-s + ⋯
L(s)  = 1  + (0.994 − 0.100i)2-s + (0.557 − 0.149i)3-s + (0.979 − 0.200i)4-s + (−1.05 − 0.283i)5-s + (0.539 − 0.204i)6-s + (0.949 + 0.313i)7-s + (0.954 − 0.297i)8-s + (0.288 − 0.166i)9-s + (−1.08 − 0.175i)10-s + (0.227 + 0.848i)11-s + (0.516 − 0.258i)12-s + (−0.721 − 0.721i)13-s + (0.976 + 0.215i)14-s − 0.632·15-s + (0.919 − 0.392i)16-s + (−0.372 + 0.644i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.931 + 0.362i$
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: \((2,\ 336,\ (\ :1/2),\ 0.931 + 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.55219 - 0.479404i\)
\(L(\frac12)\) \(\approx\) \(2.55219 - 0.479404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.142i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (-2.51 - 0.828i)T \)
good5 \( 1 + (2.36 + 0.634i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.753 - 2.81i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.60 + 2.60i)T + 13iT^{2} \)
17 \( 1 + (1.53 - 2.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.06 + 7.72i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (6.39 - 3.69i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.41 - 2.41i)T + 29iT^{2} \)
31 \( 1 + (3.59 - 6.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.63 + 1.24i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (5.51 - 5.51i)T - 43iT^{2} \)
47 \( 1 + (3.53 + 6.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.221 + 0.827i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.440 + 1.64i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.335 - 1.25i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-13.9 + 3.74i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 15.2iT - 71T^{2} \)
73 \( 1 + (-5.27 - 3.04i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.28 + 2.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.29 + 4.29i)T + 83iT^{2} \)
89 \( 1 + (-6.94 + 4.01i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.05T + 97T^{2} \)
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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

−11.78690343576747326369667488811, −10.92634729050625369196015031109, −9.727512091989984814035394738910, −8.359224357336291273379843063618, −7.65965274259013192216318106587, −6.79840612067403616599953735741, −5.13314829578075236878426833884, −4.48852177399717973812401538557, −3.26735694961934228984492198367, −1.86833602602994109476959257621, 2.12712335692966982714938289640, 3.69429602849555939560461222629, 4.22102100535691520933582228165, 5.51226638051566660548079296338, 6.89978023574201951390502516776, 7.79138081846353217027946242063, 8.355343018862773408443179266687, 9.967142801395771796718852984136, 11.01901714508718415453553663832, 11.77016079927450485542581155476

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