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Properties

Label 2-336-112.109-c1-0-6
Degree $2$
Conductor $336$
Sign $-0.706 - 0.707i$
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  + (0.762 + 1.19i)2-s + (−0.965 + 0.258i)3-s + (−0.836 + 1.81i)4-s + (2.91 + 0.780i)5-s + (−1.04 − 0.952i)6-s + (−2.27 + 1.34i)7-s + (−2.80 + 0.390i)8-s + (0.866 − 0.499i)9-s + (1.29 + 4.06i)10-s + (0.419 + 1.56i)11-s + (0.337 − 1.97i)12-s + (0.849 + 0.849i)13-s + (−3.33 − 1.68i)14-s − 3.01·15-s + (−2.60 − 3.03i)16-s + (0.359 − 0.622i)17-s + ⋯
L(s)  = 1  + (0.539 + 0.842i)2-s + (−0.557 + 0.149i)3-s + (−0.418 + 0.908i)4-s + (1.30 + 0.348i)5-s + (−0.426 − 0.388i)6-s + (−0.861 + 0.508i)7-s + (−0.990 + 0.138i)8-s + (0.288 − 0.166i)9-s + (0.408 + 1.28i)10-s + (0.126 + 0.471i)11-s + (0.0973 − 0.569i)12-s + (0.235 + 0.235i)13-s + (−0.892 − 0.451i)14-s − 0.778·15-s + (−0.650 − 0.759i)16-s + (0.0871 − 0.150i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.706 - 0.707i$
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: \((2,\ 336,\ (\ :1/2),\ -0.706 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.569226 + 1.37361i\)
\(L(\frac12)\) \(\approx\) \(0.569226 + 1.37361i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.762 - 1.19i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (2.27 - 1.34i)T \)
good5 \( 1 + (-2.91 - 0.780i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.419 - 1.56i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.849 - 0.849i)T + 13iT^{2} \)
17 \( 1 + (-0.359 + 0.622i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.07 - 4.02i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.760 + 0.439i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.65 + 3.65i)T + 29iT^{2} \)
31 \( 1 + (-1.57 + 2.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.29 - 1.41i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.90iT - 41T^{2} \)
43 \( 1 + (-8.88 + 8.88i)T - 43iT^{2} \)
47 \( 1 + (-6.46 - 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.337 + 1.26i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.89 + 10.8i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.71 + 6.41i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-13.2 + 3.55i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.61iT - 71T^{2} \)
73 \( 1 + (-12.1 - 6.99i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.793 + 1.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.16 - 2.16i)T + 83iT^{2} \)
89 \( 1 + (9.85 - 5.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.16T + 97T^{2} \)
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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

−12.23705793561961241828774966408, −11.01504211388652324822081999837, −9.662927194090998132665862297109, −9.433461062392374854481263942376, −7.920363496223158311323321964377, −6.59062478474735930935005279364, −6.14018764849070216361154244727, −5.33246094334454298382802192815, −3.95841080243151873202867314183, −2.45712314211218084293106292176, 0.998456193561930457522121302487, 2.56442565642129746622453520304, 3.99004574146944399516061709999, 5.34962898490383233008838285482, 5.99151319390887041536146635614, 6.95115398436555350877652886213, 8.887089822709450784968376577842, 9.583764446957750879807063219139, 10.47036913383728160328046222461, 11.07361327334972534332697513158

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