Properties

Label 2-392-392.109-c1-0-32
Degree $2$
Conductor $392$
Sign $-0.235 + 0.971i$
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  + (−1.41 − 0.0853i)2-s + (−0.290 − 1.92i)3-s + (1.98 + 0.240i)4-s + (−1.10 − 0.434i)5-s + (0.245 + 2.74i)6-s + (1.77 + 1.96i)7-s + (−2.78 − 0.509i)8-s + (−0.752 + 0.232i)9-s + (1.52 + 0.708i)10-s + (1.15 − 3.74i)11-s + (−0.112 − 3.89i)12-s + (4.65 − 1.06i)13-s + (−2.33 − 2.92i)14-s + (−0.515 + 2.25i)15-s + (3.88 + 0.956i)16-s + (2.11 − 1.44i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0603i)2-s + (−0.167 − 1.11i)3-s + (0.992 + 0.120i)4-s + (−0.495 − 0.194i)5-s + (0.100 + 1.11i)6-s + (0.671 + 0.741i)7-s + (−0.983 − 0.180i)8-s + (−0.250 + 0.0773i)9-s + (0.482 + 0.224i)10-s + (0.348 − 1.12i)11-s + (−0.0324 − 1.12i)12-s + (1.29 − 0.294i)13-s + (−0.625 − 0.780i)14-s + (−0.133 + 0.583i)15-s + (0.970 + 0.239i)16-s + (0.512 − 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.235 + 0.971i$
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: \((2,\ 392,\ (\ :1/2),\ -0.235 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.513719 - 0.653282i\)
\(L(\frac12)\) \(\approx\) \(0.513719 - 0.653282i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0853i)T \)
7 \( 1 + (-1.77 - 1.96i)T \)
good3 \( 1 + (0.290 + 1.92i)T + (-2.86 + 0.884i)T^{2} \)
5 \( 1 + (1.10 + 0.434i)T + (3.66 + 3.40i)T^{2} \)
11 \( 1 + (-1.15 + 3.74i)T + (-9.08 - 6.19i)T^{2} \)
13 \( 1 + (-4.65 + 1.06i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-2.11 + 1.44i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (3.04 - 1.75i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.64 + 2.48i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (2.00 - 4.16i)T + (-18.0 - 22.6i)T^{2} \)
31 \( 1 + (-1.49 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.67 + 0.125i)T + (36.5 - 5.51i)T^{2} \)
41 \( 1 + (-1.46 - 1.83i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (6.23 + 4.97i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-8.15 + 7.56i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (0.517 + 0.0387i)T + (52.4 + 7.89i)T^{2} \)
59 \( 1 + (-9.32 + 3.66i)T + (43.2 - 40.1i)T^{2} \)
61 \( 1 + (-3.74 + 0.280i)T + (60.3 - 9.09i)T^{2} \)
67 \( 1 + (9.58 + 5.53i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.89 - 3.80i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-1.56 - 1.45i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (-6.40 - 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.80 + 1.32i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (-7.80 + 2.40i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−11.24007575988053538656498803575, −10.20295931842682347995273616081, −8.725348933938717494380973716244, −8.367933397281360952068023689167, −7.58304438542182465370951827725, −6.36783047083279749070499121871, −5.76833127595901689768800577245, −3.69480726296304295256799738379, −2.09043725860805543391642801635, −0.845643593017996255347525995849, 1.62980535992684642746428837650, 3.69110575979726194355039256363, 4.44948593239185697962254903065, 5.94308048184311351541772585088, 7.16603329351490071374734564978, 7.918292713520362415241548557984, 8.962675339410353092937393950673, 9.882463311915726473628250098274, 10.51619529339941542668735872903, 11.24230208606101282224966947519

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