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Properties

Label 2-336-21.5-c1-0-7
Degree $2$
Conductor $336$
Sign $-0.340 + 0.940i$
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  + (−1.71 − 0.247i)3-s + (−1.28 + 2.23i)5-s + (0.203 − 2.63i)7-s + (2.87 + 0.849i)9-s + (−1.43 + 0.826i)11-s − 5.71i·13-s + (2.76 − 3.50i)15-s + (−3.79 − 6.56i)17-s + (−2.58 − 1.49i)19-s + (−1.00 + 4.47i)21-s + (−0.249 − 0.143i)23-s + (−0.825 − 1.43i)25-s + (−4.72 − 2.16i)27-s − 2.05i·29-s + (5.21 − 3.00i)31-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)3-s + (−0.576 + 0.998i)5-s + (0.0768 − 0.997i)7-s + (0.959 + 0.283i)9-s + (−0.431 + 0.249i)11-s − 1.58i·13-s + (0.713 − 0.906i)15-s + (−0.919 − 1.59i)17-s + (−0.594 − 0.343i)19-s + (−0.218 + 0.975i)21-s + (−0.0519 − 0.0300i)23-s + (−0.165 − 0.286i)25-s + (−0.908 − 0.417i)27-s − 0.382i·29-s + (0.936 − 0.540i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.340 + 0.940i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.340 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.298031 - 0.424775i\)
\(L(\frac12)\) \(\approx\) \(0.298031 - 0.424775i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.247i)T \)
7 \( 1 + (-0.203 + 2.63i)T \)
good5 \( 1 + (1.28 - 2.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.43 - 0.826i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.71iT - 13T^{2} \)
17 \( 1 + (3.79 + 6.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.58 + 1.49i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.249 + 0.143i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.05iT - 29T^{2} \)
31 \( 1 + (-5.21 + 3.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.877 - 1.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.28T + 41T^{2} \)
43 \( 1 + 2.46T + 43T^{2} \)
47 \( 1 + (0.186 - 0.323i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.73 + 3.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.89 - 8.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.889 - 0.513i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.18 - 2.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + (3.30 - 1.90i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.56 - 7.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.65T + 83T^{2} \)
89 \( 1 + (7.25 - 12.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.43iT - 97T^{2} \)
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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

−11.15991444158966571747683969393, −10.55388865373928860501196450289, −9.878827095681360185077198554248, −8.084414866485772857316186549054, −7.22957120670664213153124449574, −6.66930395429974060767042098821, −5.28524573265886313781737532483, −4.24727655174244728373855025945, −2.80991057042183285128581666437, −0.40899089889367479778663284974, 1.78492664648249292658765246270, 4.06445137388215299026409577301, 4.82541547100784576816787901012, 5.92974577836550975265984875506, 6.79767715010268127192566601734, 8.393535373930653808480382131015, 8.828361022634233497811527458369, 10.08697648702122049970994543601, 11.14214244405160153665502839864, 11.86755765721120124920275732567

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