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Properties

Label 2-336-7.5-c2-0-10
Degree $2$
Conductor $336$
Sign $0.687 + 0.726i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
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 + 0.866i)3-s + (3.18 + 1.83i)5-s + (−2.47 − 6.54i)7-s + (1.5 − 2.59i)9-s + (−2.59 − 4.49i)11-s − 2.95i·13-s − 6.36·15-s + (17.1 − 9.88i)17-s + (−0.805 − 0.465i)19-s + (9.37 + 7.68i)21-s + (1.58 − 2.74i)23-s + (−5.75 − 9.96i)25-s + 5.19i·27-s + 31.5·29-s + (35.9 − 20.7i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (0.636 + 0.367i)5-s + (−0.353 − 0.935i)7-s + (0.166 − 0.288i)9-s + (−0.235 − 0.408i)11-s − 0.227i·13-s − 0.424·15-s + (1.00 − 0.581i)17-s + (−0.0423 − 0.0244i)19-s + (0.446 + 0.365i)21-s + (0.0688 − 0.119i)23-s + (−0.230 − 0.398i)25-s + 0.192i·27-s + 1.08·29-s + (1.15 − 0.669i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.687 + 0.726i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.687 + 0.726i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.28156 - 0.551711i\)
\(L(\frac12)\) \(\approx\) \(1.28156 - 0.551711i\)
\(L(2)\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (2.47 + 6.54i)T \)
good5 \( 1 + (-3.18 - 1.83i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.59 + 4.49i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 2.95iT - 169T^{2} \)
17 \( 1 + (-17.1 + 9.88i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (0.805 + 0.465i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-1.58 + 2.74i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 31.5T + 841T^{2} \)
31 \( 1 + (-35.9 + 20.7i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (5.95 - 10.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 60.5iT - 1.68e3T^{2} \)
43 \( 1 - 68.8T + 1.84e3T^{2} \)
47 \( 1 + (20.8 + 12.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (33.1 + 57.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-79.4 + 45.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (10.2 + 5.90i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-46.3 - 80.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 61.0T + 5.04e3T^{2} \)
73 \( 1 + (119. - 68.8i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (63.7 - 110. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 24.2iT - 6.88e3T^{2} \)
89 \( 1 + (-24.3 - 14.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 49.2iT - 9.40e3T^{2} \)
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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

−11.06724369303941609647678209937, −10.13168057051144499322007206326, −9.843974547028368181753182892654, −8.373650844955248661389956887944, −7.22465217128194577738639083598, −6.30473978819706849149966495115, −5.36205496156493557079902793216, −4.08526451707922203931100116275, −2.79601343584967892908991889794, −0.75916986338918795648731732342, 1.44515994520482943574094788247, 2.86357657159201621856459949036, 4.66042540531379368949290524697, 5.69423065296058188916955660530, 6.35297792021819115258403937195, 7.64900233417439069572009309440, 8.720516168170087018616183902653, 9.672254404083383993456070567270, 10.41557417248755197525116509568, 11.67448092690289934474748679086

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