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Properties

Label 2-336-112.37-c1-0-14
Degree $2$
Conductor $336$
Sign $-0.203 + 0.979i$
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  + (−0.218 − 1.39i)2-s + (−0.965 − 0.258i)3-s + (−1.90 + 0.611i)4-s + (2.14 − 0.575i)5-s + (−0.150 + 1.40i)6-s + (2.53 − 0.752i)7-s + (1.27 + 2.52i)8-s + (0.866 + 0.499i)9-s + (−1.27 − 2.87i)10-s + (−0.347 + 1.29i)11-s + (1.99 − 0.0980i)12-s + (3.37 − 3.37i)13-s + (−1.60 − 3.37i)14-s − 2.22·15-s + (3.25 − 2.32i)16-s + (−2.18 − 3.77i)17-s + ⋯
L(s)  = 1  + (−0.154 − 0.987i)2-s + (−0.557 − 0.149i)3-s + (−0.952 + 0.305i)4-s + (0.959 − 0.257i)5-s + (−0.0612 + 0.574i)6-s + (0.958 − 0.284i)7-s + (0.449 + 0.893i)8-s + (0.288 + 0.166i)9-s + (−0.402 − 0.908i)10-s + (−0.104 + 0.391i)11-s + (0.576 − 0.0283i)12-s + (0.935 − 0.935i)13-s + (−0.429 − 0.903i)14-s − 0.573·15-s + (0.812 − 0.582i)16-s + (−0.528 − 0.916i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.203 + 0.979i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.203 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.757490 - 0.930813i\)
\(L(\frac12)\) \(\approx\) \(0.757490 - 0.930813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.218 + 1.39i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (-2.53 + 0.752i)T \)
good5 \( 1 + (-2.14 + 0.575i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.347 - 1.29i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.37 + 3.37i)T - 13iT^{2} \)
17 \( 1 + (2.18 + 3.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.793 - 2.96i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (6.37 + 3.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.40 + 6.40i)T - 29iT^{2} \)
31 \( 1 + (-1.28 - 2.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.49 + 0.935i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.86iT - 41T^{2} \)
43 \( 1 + (3.51 + 3.51i)T + 43iT^{2} \)
47 \( 1 + (1.61 - 2.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.350 - 1.30i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.0438 - 0.163i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.40 - 12.7i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-9.54 - 2.55i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.29iT - 71T^{2} \)
73 \( 1 + (-0.678 + 0.391i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.20 - 9.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.56 - 1.56i)T - 83iT^{2} \)
89 \( 1 + (13.1 + 7.60i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.11T + 97T^{2} \)
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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

−11.32860645294211152061327986517, −10.31170314731186187090332839773, −9.879187013666020709594625949171, −8.546913170093260804064916554991, −7.80259407367117901834748714877, −6.15400850934167111782816832500, −5.17202913144096621043343813366, −4.20922966005739846177877639134, −2.39971076979681842629352510984, −1.13981722949983406435334623454, 1.68285199463370798350691457352, 4.05568465002703855698177835867, 5.18734065821111264424136446306, 6.05784393738491448151943350934, 6.71385802472838815248700860832, 8.114165706134014874631093895491, 8.875955293062258752302442276096, 9.871463751278645622701265271918, 10.78508953748847597781384165671, 11.64514457280061642256956353137

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