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Properties

Label 2-336-7.2-c3-0-23
Degree $2$
Conductor $336$
Sign $-0.701 - 0.712i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 2.59i)3-s + (7.5 − 12.9i)5-s + (−17.5 − 6.06i)7-s + (−4.5 + 7.79i)9-s + (−4.5 − 7.79i)11-s − 88·13-s − 45·15-s + (42 + 72.7i)17-s + (52 − 90.0i)19-s + (10.5 + 54.5i)21-s + (−42 + 72.7i)23-s + (−50 − 86.6i)25-s + 27·27-s + 51·29-s + (92.5 + 160. i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.670 − 1.16i)5-s + (−0.944 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.123 − 0.213i)11-s − 1.87·13-s − 0.774·15-s + (0.599 + 1.03i)17-s + (0.627 − 1.08i)19-s + (0.109 + 0.566i)21-s + (−0.380 + 0.659i)23-s + (−0.400 − 0.692i)25-s + 0.192·27-s + 0.326·29-s + (0.535 + 0.928i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2664203726\)
\(L(\frac12)\) \(\approx\) \(0.2664203726\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 + (17.5 + 6.06i)T \)
good5 \( 1 + (-7.5 + 12.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (4.5 + 7.79i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 88T + 2.19e3T^{2} \)
17 \( 1 + (-42 - 72.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-52 + 90.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (42 - 72.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 51T + 2.43e4T^{2} \)
31 \( 1 + (-92.5 - 160. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (22 - 38.1i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 168T + 6.89e4T^{2} \)
43 \( 1 + 326T + 7.95e4T^{2} \)
47 \( 1 + (69 - 119. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (319.5 + 553. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-79.5 - 137. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (361 - 625. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (83 + 143. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.08e3T + 3.57e5T^{2} \)
73 \( 1 + (109 + 188. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (291.5 - 504. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 597T + 5.71e5T^{2} \)
89 \( 1 + (-519 + 898. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 169T + 9.12e5T^{2} \)
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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

−10.25576065398391225757010550500, −9.720259096452749960556992154629, −8.744761085666186099424341067458, −7.59281829393495495719420288448, −6.63525632926821996004962681336, −5.52116745542852503046027327156, −4.73722222220035555290693618091, −2.99703763955553344261495798388, −1.47184416633862354614292300231, −0.094817040078483643258617881656, 2.42983144822964596863136607039, 3.24117886410597422240745604571, 4.86765044910558577603093834372, 5.91382134062253626047663422255, 6.78905655987627756655892213963, 7.71331329562421992596159268103, 9.424134482519009419504758690818, 9.944000680502086649920922572583, 10.39757026642541822150939783551, 11.83391316897089071891655080036

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