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Properties

Label 2-392-7.4-c3-0-29
Degree $2$
Conductor $392$
Sign $-0.605 - 0.795i$
Analytic cond. $23.1287$
Root an. cond. $4.80923$
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  + (4.27 − 7.40i)3-s + (−9.27 − 16.0i)5-s + (−23.0 − 39.9i)9-s + (−31.6 + 54.8i)11-s − 16.7·13-s − 158.·15-s + (20.6 − 35.7i)17-s + (−19.5 − 33.9i)19-s + (10.9 + 18.8i)23-s + (−109. + 189. i)25-s − 163.·27-s + 138.·29-s + (47.8 − 82.8i)31-s + (270. + 468. i)33-s + (−88.2 − 152. i)37-s + ⋯
L(s)  = 1  + (0.822 − 1.42i)3-s + (−0.829 − 1.43i)5-s + (−0.853 − 1.47i)9-s + (−0.867 + 1.50i)11-s − 0.357·13-s − 2.72·15-s + (0.294 − 0.510i)17-s + (−0.236 − 0.409i)19-s + (0.0988 + 0.171i)23-s + (−0.876 + 1.51i)25-s − 1.16·27-s + 0.884·29-s + (0.277 − 0.480i)31-s + (1.42 + 2.47i)33-s + (−0.392 − 0.679i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(23.1287\)
Root analytic conductor: \(4.80923\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :3/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9418070212\)
\(L(\frac12)\) \(\approx\) \(0.9418070212\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-4.27 + 7.40i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (9.27 + 16.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (31.6 - 54.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 16.7T + 2.19e3T^{2} \)
17 \( 1 + (-20.6 + 35.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (19.5 + 33.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-10.9 - 18.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 138.T + 2.43e4T^{2} \)
31 \( 1 + (-47.8 + 82.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (88.2 + 152. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 407.T + 6.89e4T^{2} \)
43 \( 1 - 100.T + 7.95e4T^{2} \)
47 \( 1 + (-72.1 - 124. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-204. + 354. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-0.426 + 0.737i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-203. - 352. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-4.69 + 8.13i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 944.T + 3.57e5T^{2} \)
73 \( 1 + (43.1 - 74.6i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (281. + 488. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 969.T + 5.71e5T^{2} \)
89 \( 1 + (752. + 1.30e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 956.T + 9.12e5T^{2} \)
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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

−10.07939948977765443674193940770, −9.023678240790652318459350209076, −8.286837962544351937257456219627, −7.55963242722847808595326282139, −6.96457013750317730283328492855, −5.27623167171161002252570236704, −4.33354916615478539059416982160, −2.70673851432570035477312020839, −1.54773632445159211183274262944, −0.27896322077059292581206016119, 2.82159276028537658453531702698, 3.27291697695442013131509125824, 4.25414114760691592391605586840, 5.56997650052164356165308970750, 6.87293005418972963442414619027, 8.153433352697179960408800539855, 8.497460209255130355768742597114, 9.950171919046361201530308529004, 10.52920172551105518823306517634, 11.02527564897174476099331826123

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