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Properties

Label 2-336-7.4-c3-0-7
Degree $2$
Conductor $336$
Sign $-0.723 - 0.690i$
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 + (6.41 + 11.1i)5-s + (−6.32 − 17.4i)7-s + (−4.5 − 7.79i)9-s + (−18.4 + 31.8i)11-s + 87.1·13-s − 38.4·15-s + (−51.2 + 88.8i)17-s + (47.9 + 82.9i)19-s + (54.7 + 9.67i)21-s + (−48 − 83.1i)23-s + (−19.7 + 34.1i)25-s + 27·27-s − 212.·29-s + (−79.6 + 137. i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.573 + 0.993i)5-s + (−0.341 − 0.939i)7-s + (−0.166 − 0.288i)9-s + (−0.504 + 0.874i)11-s + 1.85·13-s − 0.662·15-s + (−0.731 + 1.26i)17-s + (0.578 + 1.00i)19-s + (0.568 + 0.100i)21-s + (−0.435 − 0.753i)23-s + (−0.157 + 0.273i)25-s + 0.192·27-s − 1.35·29-s + (−0.461 + 0.799i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.723 - 0.690i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.723 - 0.690i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.305471246\)
\(L(\frac12)\) \(\approx\) \(1.305471246\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.32 + 17.4i)T \)
good5 \( 1 + (-6.41 - 11.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (18.4 - 31.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 87.1T + 2.19e3T^{2} \)
17 \( 1 + (51.2 - 88.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-47.9 - 82.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (48 + 83.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 212.T + 2.43e4T^{2} \)
31 \( 1 + (79.6 - 137. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (64.3 + 111. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 298.T + 6.89e4T^{2} \)
43 \( 1 - 33.3T + 7.95e4T^{2} \)
47 \( 1 + (-135. - 234. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (224. - 388. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (334. - 578. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-121. - 211. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (167. - 290. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 339.T + 3.57e5T^{2} \)
73 \( 1 + (-459. + 795. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (68.1 + 118. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 287.T + 5.71e5T^{2} \)
89 \( 1 + (80.9 + 140. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 182.T + 9.12e5T^{2} \)
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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

−10.94165759103291523410452161410, −10.63279769766752459547072893776, −9.976659369470587896299292395135, −8.781491239773782237620542673124, −7.54239058197364302563797606759, −6.47971324926329654445316712235, −5.83662219045582266930403433230, −4.21380850001322398727301244247, −3.38671547859138471078503852012, −1.68281005928335084044082621519, 0.47729186977758951895913446369, 1.87237203208634086000365996236, 3.34339723972194938524730313630, 5.17701630209141828746995074457, 5.69485935642311279206395016642, 6.71007503630590430637969112669, 8.140424600509647120335409737876, 8.940981403016665204988958978149, 9.527010974028386451477830595694, 11.18530322706311339855899412407

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