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Properties

Label 2-336-16.13-c1-0-4
Degree $2$
Conductor $336$
Sign $0.516 - 0.856i$
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  + (−1.32 + 0.491i)2-s + (−0.707 − 0.707i)3-s + (1.51 − 1.30i)4-s + (−1.17 + 1.17i)5-s + (1.28 + 0.589i)6-s − i·7-s + (−1.36 + 2.47i)8-s + 1.00i·9-s + (0.978 − 2.13i)10-s + (−0.961 + 0.961i)11-s + (−1.99 − 0.150i)12-s + (2.26 + 2.26i)13-s + (0.491 + 1.32i)14-s + 1.65·15-s + (0.599 − 3.95i)16-s + 3.43·17-s + ⋯
L(s)  = 1  + (−0.937 + 0.347i)2-s + (−0.408 − 0.408i)3-s + (0.758 − 0.651i)4-s + (−0.524 + 0.524i)5-s + (0.524 + 0.240i)6-s − 0.377i·7-s + (−0.484 + 0.874i)8-s + 0.333i·9-s + (0.309 − 0.674i)10-s + (−0.289 + 0.289i)11-s + (−0.575 − 0.0433i)12-s + (0.626 + 0.626i)13-s + (0.131 + 0.354i)14-s + 0.428·15-s + (0.149 − 0.988i)16-s + 0.832·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.516 - 0.856i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.516 - 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.569564 + 0.321461i\)
\(L(\frac12)\) \(\approx\) \(0.569564 + 0.321461i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.491i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + iT \)
good5 \( 1 + (1.17 - 1.17i)T - 5iT^{2} \)
11 \( 1 + (0.961 - 0.961i)T - 11iT^{2} \)
13 \( 1 + (-2.26 - 2.26i)T + 13iT^{2} \)
17 \( 1 - 3.43T + 17T^{2} \)
19 \( 1 + (-0.568 - 0.568i)T + 19iT^{2} \)
23 \( 1 - 2.04iT - 23T^{2} \)
29 \( 1 + (-6.38 - 6.38i)T + 29iT^{2} \)
31 \( 1 - 9.65T + 31T^{2} \)
37 \( 1 + (8.01 - 8.01i)T - 37iT^{2} \)
41 \( 1 + 0.877iT - 41T^{2} \)
43 \( 1 + (-1.46 + 1.46i)T - 43iT^{2} \)
47 \( 1 + 0.509T + 47T^{2} \)
53 \( 1 + (4.42 - 4.42i)T - 53iT^{2} \)
59 \( 1 + (-1.96 + 1.96i)T - 59iT^{2} \)
61 \( 1 + (-4.61 - 4.61i)T + 61iT^{2} \)
67 \( 1 + (-0.0452 - 0.0452i)T + 67iT^{2} \)
71 \( 1 + 9.10iT - 71T^{2} \)
73 \( 1 + 9.52iT - 73T^{2} \)
79 \( 1 - 4.83T + 79T^{2} \)
83 \( 1 + (9.73 + 9.73i)T + 83iT^{2} \)
89 \( 1 + 12.5iT - 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−11.59189504846183159805861444176, −10.64006148432387355737210066538, −10.02616231088807762863944613627, −8.733084947858842543949557090046, −7.81721755866126009385579871038, −7.03010900356556714443937242917, −6.27465206663513928950870343486, −4.95880062217571726969108402039, −3.16972166193428653454854015398, −1.35168992143907027914991541249, 0.75058173803746508909755794102, 2.83205231553231726059460341573, 4.12963842443028441272278325651, 5.54969475223967810216655553975, 6.65209266543572368559042988493, 8.135641573010501418956081087974, 8.418469938568694308634625784318, 9.696598929534033365958647919934, 10.39700536567264155740021533791, 11.32387017678077334614782122631

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