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Properties

Label 2-336-48.11-c1-0-33
Degree $2$
Conductor $336$
Sign $0.281 + 0.959i$
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  + (−1.31 − 0.508i)2-s + (1.45 − 0.945i)3-s + (1.48 + 1.34i)4-s + (1.69 − 1.69i)5-s + (−2.39 + 0.510i)6-s + 7-s + (−1.27 − 2.52i)8-s + (1.21 − 2.74i)9-s + (−3.09 + 1.37i)10-s + (0.445 + 0.445i)11-s + (3.42 + 0.543i)12-s + (−1.51 + 1.51i)13-s + (−1.31 − 0.508i)14-s + (0.856 − 4.06i)15-s + (0.401 + 3.97i)16-s − 1.39i·17-s + ⋯
L(s)  = 1  + (−0.933 − 0.359i)2-s + (0.837 − 0.546i)3-s + (0.741 + 0.670i)4-s + (0.758 − 0.758i)5-s + (−0.978 + 0.208i)6-s + 0.377·7-s + (−0.451 − 0.892i)8-s + (0.403 − 0.914i)9-s + (−0.980 + 0.435i)10-s + (0.134 + 0.134i)11-s + (0.987 + 0.156i)12-s + (−0.419 + 0.419i)13-s + (−0.352 − 0.135i)14-s + (0.221 − 1.04i)15-s + (0.100 + 0.994i)16-s − 0.337i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.281 + 0.959i$
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: \((2,\ 336,\ (\ :1/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06612 - 0.798132i\)
\(L(\frac12)\) \(\approx\) \(1.06612 - 0.798132i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 + 0.508i)T \)
3 \( 1 + (-1.45 + 0.945i)T \)
7 \( 1 - T \)
good5 \( 1 + (-1.69 + 1.69i)T - 5iT^{2} \)
11 \( 1 + (-0.445 - 0.445i)T + 11iT^{2} \)
13 \( 1 + (1.51 - 1.51i)T - 13iT^{2} \)
17 \( 1 + 1.39iT - 17T^{2} \)
19 \( 1 + (-0.965 - 0.965i)T + 19iT^{2} \)
23 \( 1 - 6.04iT - 23T^{2} \)
29 \( 1 + (4.93 + 4.93i)T + 29iT^{2} \)
31 \( 1 + 0.470iT - 31T^{2} \)
37 \( 1 + (-1.47 - 1.47i)T + 37iT^{2} \)
41 \( 1 + 8.35T + 41T^{2} \)
43 \( 1 + (-8.97 + 8.97i)T - 43iT^{2} \)
47 \( 1 + 4.84T + 47T^{2} \)
53 \( 1 + (-3.42 + 3.42i)T - 53iT^{2} \)
59 \( 1 + (-6.61 - 6.61i)T + 59iT^{2} \)
61 \( 1 + (8.04 - 8.04i)T - 61iT^{2} \)
67 \( 1 + (-4.38 - 4.38i)T + 67iT^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 + 14.0iT - 73T^{2} \)
79 \( 1 - 16.5iT - 79T^{2} \)
83 \( 1 + (9.24 - 9.24i)T - 83iT^{2} \)
89 \( 1 - 5.25T + 89T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−11.48792585425000671655793145769, −10.01463460759898881074972870703, −9.406785962497363346484259722211, −8.738181338870705402798342009124, −7.74574510851109713933631994053, −6.97077479220189182240770013796, −5.59181794503723276341434479789, −3.87586845618307784367976760440, −2.34401411600685674300790605700, −1.36077726907402234096344085931, 1.98018337187501363217716523632, 3.05856650408648719142148886941, 4.88374671391598887241561181762, 6.11579347824094062287021096373, 7.18521613929933215544529583224, 8.110427844461694541774457034111, 8.992572652896669219069179579993, 9.843033943096939952238506777019, 10.52418962108727677939237733895, 11.18586457859610957899559220405

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