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Properties

Label 2-336-112.109-c1-0-23
Degree $2$
Conductor $336$
Sign $-0.839 + 0.543i$
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  + (−0.120 − 1.40i)2-s + (−0.965 + 0.258i)3-s + (−1.97 + 0.340i)4-s + (0.433 + 0.116i)5-s + (0.481 + 1.32i)6-s + (2.04 − 1.68i)7-s + (0.717 + 2.73i)8-s + (0.866 − 0.499i)9-s + (0.111 − 0.624i)10-s + (−1.18 − 4.41i)11-s + (1.81 − 0.839i)12-s + (−3.32 − 3.32i)13-s + (−2.61 − 2.67i)14-s − 0.448·15-s + (3.76 − 1.34i)16-s + (−2.68 + 4.64i)17-s + ⋯
L(s)  = 1  + (−0.0854 − 0.996i)2-s + (−0.557 + 0.149i)3-s + (−0.985 + 0.170i)4-s + (0.193 + 0.0519i)5-s + (0.196 + 0.542i)6-s + (0.772 − 0.635i)7-s + (0.253 + 0.967i)8-s + (0.288 − 0.166i)9-s + (0.0351 − 0.197i)10-s + (−0.356 − 1.33i)11-s + (0.524 − 0.242i)12-s + (−0.920 − 0.920i)13-s + (−0.699 − 0.714i)14-s − 0.115·15-s + (0.942 − 0.335i)16-s + (−0.650 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.839 + 0.543i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.839 + 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.237257 - 0.802511i\)
\(L(\frac12)\) \(\approx\) \(0.237257 - 0.802511i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.120 + 1.40i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-2.04 + 1.68i)T \)
good5 \( 1 + (-0.433 - 0.116i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.18 + 4.41i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (3.32 + 3.32i)T + 13iT^{2} \)
17 \( 1 + (2.68 - 4.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.47 + 5.51i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.0663 + 0.0382i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.56 + 3.56i)T + 29iT^{2} \)
31 \( 1 + (-0.347 + 0.602i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-11.1 - 2.97i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.69iT - 41T^{2} \)
43 \( 1 + (-6.59 + 6.59i)T - 43iT^{2} \)
47 \( 1 + (0.711 + 1.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.86 + 6.94i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.230 + 0.860i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.37 - 12.5i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-4.23 + 1.13i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.19iT - 71T^{2} \)
73 \( 1 + (-8.04 - 4.64i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.89 + 5.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.19 - 4.19i)T + 83iT^{2} \)
89 \( 1 + (-6.23 + 3.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−11.13270971194826026898431086084, −10.51368218395956949457739891818, −9.649419544760731603210818548253, −8.426401314286828691891922803164, −7.64657214396701450511091712849, −5.98284035224023709131918151970, −4.99390568613459856830100523449, −3.96221727723981812826816316715, −2.49746552846729546095487713553, −0.66247701591985212723293225393, 1.97350597341157077312851171635, 4.42424463686843139781650709255, 5.09302762223236792259680677148, 6.07981264936972302062803096321, 7.30726762075309960102349393643, 7.75865882878311653606911291320, 9.304183307543315800997459623290, 9.656781150978744380160869759837, 11.04757528285902067880371983212, 12.11320898707963977727521856462

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