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Properties

Label 2-392-392.109-c1-0-41
Degree $2$
Conductor $392$
Sign $-0.954 - 0.299i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
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  + (−0.634 − 1.26i)2-s + (−0.0230 − 0.152i)3-s + (−1.19 + 1.60i)4-s + (−0.525 − 0.206i)5-s + (−0.178 + 0.126i)6-s + (−2.63 − 0.225i)7-s + (2.78 + 0.494i)8-s + (2.84 − 0.877i)9-s + (0.0726 + 0.795i)10-s + (1.80 − 5.84i)11-s + (0.272 + 0.145i)12-s + (−6.52 + 1.48i)13-s + (1.38 + 3.47i)14-s + (−0.0194 + 0.0850i)15-s + (−1.14 − 3.83i)16-s + (−4.09 + 2.79i)17-s + ⋯
L(s)  = 1  + (−0.448 − 0.893i)2-s + (−0.0132 − 0.0882i)3-s + (−0.597 + 0.801i)4-s + (−0.235 − 0.0922i)5-s + (−0.0728 + 0.0514i)6-s + (−0.996 − 0.0853i)7-s + (0.984 + 0.174i)8-s + (0.947 − 0.292i)9-s + (0.0229 + 0.251i)10-s + (0.543 − 1.76i)11-s + (0.0786 + 0.0420i)12-s + (−1.81 + 0.413i)13-s + (0.370 + 0.928i)14-s + (−0.00501 + 0.0219i)15-s + (−0.285 − 0.958i)16-s + (−0.994 + 0.677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.954 - 0.299i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.954 - 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0573768 + 0.374914i\)
\(L(\frac12)\) \(\approx\) \(0.0573768 + 0.374914i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.634 + 1.26i)T \)
7 \( 1 + (2.63 + 0.225i)T \)
good3 \( 1 + (0.0230 + 0.152i)T + (-2.86 + 0.884i)T^{2} \)
5 \( 1 + (0.525 + 0.206i)T + (3.66 + 3.40i)T^{2} \)
11 \( 1 + (-1.80 + 5.84i)T + (-9.08 - 6.19i)T^{2} \)
13 \( 1 + (6.52 - 1.48i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (4.09 - 2.79i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (3.64 - 2.10i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.89 + 1.29i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (2.65 - 5.51i)T + (-18.0 - 22.6i)T^{2} \)
31 \( 1 + (-3.34 + 5.78i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.30 + 0.397i)T + (36.5 - 5.51i)T^{2} \)
41 \( 1 + (4.66 + 5.85i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (1.84 + 1.47i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-2.94 + 2.73i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (2.92 + 0.219i)T + (52.4 + 7.89i)T^{2} \)
59 \( 1 + (-0.729 + 0.286i)T + (43.2 - 40.1i)T^{2} \)
61 \( 1 + (1.54 - 0.115i)T + (60.3 - 9.09i)T^{2} \)
67 \( 1 + (0.598 + 0.345i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.58 - 1.24i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-5.90 - 5.47i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (5.41 + 9.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.4 - 2.38i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (-9.73 + 3.00i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 + 8.51T + 97T^{2} \)
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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

−10.70321051044692352095094371557, −9.941078456317295364777555519387, −9.161110912899224628053887675012, −8.281418354898254157106357123834, −7.13110120665328038232265877364, −6.16110876782114394674600951560, −4.37953901687630680311023468535, −3.61915568712580118321849092772, −2.20140034137085691922461994363, −0.27650692286339943488677371305, 2.19249458010183430557279725786, 4.30367591781597401808230040032, 4.91256535049433360134221866950, 6.51909845029689473652101628589, 7.12098154416245654999108598802, 7.78061966940276963183316461246, 9.455709443401399193739802742445, 9.611320948853773958464530126508, 10.43410183301515318510595342146, 11.91457544404462609051903157883

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