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Properties

Label 2-336-112.5-c2-0-57
Degree $2$
Conductor $336$
Sign $-0.112 + 0.993i$
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  + (1.99 − 0.198i)2-s + (0.448 − 1.67i)3-s + (3.92 − 0.790i)4-s + (−2.42 − 9.06i)5-s + (0.559 − 3.41i)6-s + (6.77 + 1.75i)7-s + (7.64 − 2.35i)8-s + (−2.59 − 1.50i)9-s + (−6.63 − 17.5i)10-s + (−4.03 + 15.0i)11-s + (0.435 − 6.91i)12-s + (−8.68 − 8.68i)13-s + (13.8 + 2.14i)14-s − 16.2·15-s + (14.7 − 6.19i)16-s + (12.8 − 7.40i)17-s + ⋯
L(s)  = 1  + (0.995 − 0.0993i)2-s + (0.149 − 0.557i)3-s + (0.980 − 0.197i)4-s + (−0.485 − 1.81i)5-s + (0.0933 − 0.569i)6-s + (0.968 + 0.250i)7-s + (0.955 − 0.294i)8-s + (−0.288 − 0.166i)9-s + (−0.663 − 1.75i)10-s + (−0.366 + 1.36i)11-s + (0.0362 − 0.576i)12-s + (−0.668 − 0.668i)13-s + (0.988 + 0.153i)14-s − 1.08·15-s + (0.921 − 0.387i)16-s + (0.754 − 0.435i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.112 + 0.993i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ -0.112 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.10267 - 2.35450i\)
\(L(\frac12)\) \(\approx\) \(2.10267 - 2.35450i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 + 0.198i)T \)
3 \( 1 + (-0.448 + 1.67i)T \)
7 \( 1 + (-6.77 - 1.75i)T \)
good5 \( 1 + (2.42 + 9.06i)T + (-21.6 + 12.5i)T^{2} \)
11 \( 1 + (4.03 - 15.0i)T + (-104. - 60.5i)T^{2} \)
13 \( 1 + (8.68 + 8.68i)T + 169iT^{2} \)
17 \( 1 + (-12.8 + 7.40i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (17.5 - 4.71i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-8.25 - 4.76i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-28.7 + 28.7i)T - 841iT^{2} \)
31 \( 1 + (3.46 - 2.00i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (0.945 - 0.253i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 23.1T + 1.68e3T^{2} \)
43 \( 1 + (-21.1 - 21.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-72.7 - 42.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (5.61 - 20.9i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-18.2 - 4.88i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (106. - 28.6i)T + (3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-7.92 - 2.12i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 22.4iT - 5.04e3T^{2} \)
73 \( 1 + (-16.6 - 28.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-10.9 + 19.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (98.1 + 98.1i)T + 6.88e3iT^{2} \)
89 \( 1 + (-46.7 + 80.9i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 114. iT - 9.40e3T^{2} \)
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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

−11.63854767376167300918648143193, −10.30913702284962092271940284766, −9.092237004605242060321690561175, −7.82478749455229000930153786389, −7.59313482791588231978626254872, −5.80248401098661173223367508219, −4.87860083398574241139059160250, −4.35091559180002632251383968716, −2.39100282775219530088144949923, −1.15483131211521952497413770690, 2.44392626667956025845859026665, 3.39421564864430878938929532059, 4.34930127842844411507369099890, 5.63332861490033743680950105366, 6.70785796399345561961537088609, 7.56193974423994129599252096908, 8.484990838077491640944287217963, 10.36230406561641774620906118084, 10.82039566360470630775960632719, 11.36853501274947813155454684352

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