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Properties

Label 2-336-84.83-c2-0-19
Degree $2$
Conductor $336$
Sign $0.634 + 0.773i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.34 − 1.87i)3-s + 6.48·5-s + (4.69 − 5.19i)7-s + (2 + 8.77i)9-s + 8.12i·13-s + (−15.1 − 12.1i)15-s + 12.9·17-s + 23.4·19-s + (−20.7 + 3.41i)21-s − 30.3·23-s + 17·25-s + (11.7 − 24.3i)27-s − 52.6i·29-s + 18.7·31-s + (30.3 − 33.6i)35-s + ⋯
L(s)  = 1  + (−0.781 − 0.623i)3-s + 1.29·5-s + (0.670 − 0.742i)7-s + (0.222 + 0.974i)9-s + 0.624i·13-s + (−1.01 − 0.808i)15-s + 0.762·17-s + 1.23·19-s + (−0.986 + 0.162i)21-s − 1.32·23-s + 0.680·25-s + (0.434 − 0.900i)27-s − 1.81i·29-s + 0.605·31-s + (0.868 − 0.962i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.634 + 0.773i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.634 + 0.773i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.58746 - 0.751271i\)
\(L(\frac12)\) \(\approx\) \(1.58746 - 0.751271i\)
\(L(2)\) 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 \( 1 + (2.34 + 1.87i)T \)
7 \( 1 + (-4.69 + 5.19i)T \)
good5 \( 1 - 6.48T + 25T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 - 8.12iT - 169T^{2} \)
17 \( 1 - 12.9T + 289T^{2} \)
19 \( 1 - 23.4T + 361T^{2} \)
23 \( 1 + 30.3T + 529T^{2} \)
29 \( 1 + 52.6iT - 841T^{2} \)
31 \( 1 - 18.7T + 961T^{2} \)
37 \( 1 - 58T + 1.36e3T^{2} \)
41 \( 1 + 12.9T + 1.68e3T^{2} \)
43 \( 1 + 34.6iT - 1.84e3T^{2} \)
47 \( 1 + 22.4iT - 2.20e3T^{2} \)
53 \( 1 - 52.6iT - 2.80e3T^{2} \)
59 \( 1 + 11.2iT - 3.48e3T^{2} \)
61 \( 1 + 56.8iT - 3.72e3T^{2} \)
67 \( 1 - 55.4iT - 4.48e3T^{2} \)
71 \( 1 + 121.T + 5.04e3T^{2} \)
73 \( 1 - 32.4iT - 5.32e3T^{2} \)
79 \( 1 - 100. iT - 6.24e3T^{2} \)
83 \( 1 - 145. iT - 6.88e3T^{2} \)
89 \( 1 + 155.T + 7.92e3T^{2} \)
97 \( 1 + 113. iT - 9.40e3T^{2} \)
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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

−11.37729441485674087804256335188, −10.14206170199970658828168978419, −9.753612362645916062062064443975, −8.133833533925708545237759146227, −7.31613052095947938542089483521, −6.18370502926745726340814541060, −5.51072046969029390798467139006, −4.31378043035607099918054703250, −2.22164137050150520190032209206, −1.09211622231491916817227812541, 1.41409439778893991808422241098, 3.05728147744662978596544326643, 4.77978686761154785804977421651, 5.61170158159585835660271199392, 6.14083438555525804328858336108, 7.67509433434214206347316615118, 8.946705728523758148219245910446, 9.802552462668897742332963331733, 10.37036640844356921197722449677, 11.47812937434079206509302636037

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