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Properties

Label 2-336-21.17-c3-0-31
Degree $2$
Conductor $336$
Sign $-0.458 + 0.888i$
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  + (−5.18 − 0.262i)3-s + (−2.24 − 3.88i)5-s + (9.71 − 15.7i)7-s + (26.8 + 2.72i)9-s + (20.2 + 11.7i)11-s − 5.91i·13-s + (10.6 + 20.7i)15-s + (−58.0 + 100. i)17-s + (−8.02 + 4.63i)19-s + (−54.5 + 79.2i)21-s + (107. − 62.1i)23-s + (52.4 − 90.7i)25-s + (−138. − 21.1i)27-s − 207. i·29-s + (−122. − 70.8i)31-s + ⋯
L(s)  = 1  + (−0.998 − 0.0504i)3-s + (−0.200 − 0.347i)5-s + (0.524 − 0.851i)7-s + (0.994 + 0.100i)9-s + (0.555 + 0.320i)11-s − 0.126i·13-s + (0.183 + 0.357i)15-s + (−0.828 + 1.43i)17-s + (−0.0969 + 0.0559i)19-s + (−0.566 + 0.823i)21-s + (0.976 − 0.563i)23-s + (0.419 − 0.726i)25-s + (−0.988 − 0.150i)27-s − 1.33i·29-s + (−0.711 − 0.410i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.458 + 0.888i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.458 + 0.888i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9477816048\)
\(L(\frac12)\) \(\approx\) \(0.9477816048\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.18 + 0.262i)T \)
7 \( 1 + (-9.71 + 15.7i)T \)
good5 \( 1 + (2.24 + 3.88i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-20.2 - 11.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 5.91iT - 2.19e3T^{2} \)
17 \( 1 + (58.0 - 100. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (8.02 - 4.63i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-107. + 62.1i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 207. iT - 2.43e4T^{2} \)
31 \( 1 + (122. + 70.8i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (149. + 259. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 508.T + 6.89e4T^{2} \)
43 \( 1 + 391.T + 7.95e4T^{2} \)
47 \( 1 + (40.2 + 69.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (258. + 149. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-102. + 177. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (543. - 313. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (51.3 - 89.0i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 46.9iT - 3.57e5T^{2} \)
73 \( 1 + (228. + 131. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (533. + 924. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 270.T + 5.71e5T^{2} \)
89 \( 1 + (443. + 768. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 219. iT - 9.12e5T^{2} \)
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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

−10.87162198420023083905194461488, −10.21192772726234136153546049673, −8.950944883025278726715186530328, −7.84447658534103596741246835273, −6.87152759686833439935841277654, −5.96746734830252364589365257844, −4.62731164286983349950134425775, −4.05021010508381613529282861377, −1.73730105119271177714333323216, −0.41909662787306163045479441614, 1.38321231661532605736560019866, 3.10155796143706961788567727477, 4.69450367763794350200374504211, 5.41587663853729881864585577911, 6.63170483998945979932457262151, 7.33004164485749619522414719342, 8.818318969179638769811371616825, 9.499092303714174984280120135422, 10.99134604778690551959171972942, 11.24133933816466703513078800779

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