Properties

Label 2-392-392.205-c1-0-47
Degree $2$
Conductor $392$
Sign $-0.994 + 0.100i$
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  + (−0.0204 − 1.41i)2-s + (0.290 − 1.92i)3-s + (−1.99 + 0.0577i)4-s + (1.10 − 0.434i)5-s + (−2.72 − 0.370i)6-s + (1.77 − 1.96i)7-s + (0.122 + 2.82i)8-s + (−0.752 − 0.232i)9-s + (−0.637 − 1.55i)10-s + (−1.15 − 3.74i)11-s + (−0.468 + 3.86i)12-s + (−4.65 − 1.06i)13-s + (−2.81 − 2.47i)14-s + (−0.515 − 2.25i)15-s + (3.99 − 0.230i)16-s + (2.11 + 1.44i)17-s + ⋯
L(s)  = 1  + (−0.0144 − 0.999i)2-s + (0.167 − 1.11i)3-s + (−0.999 + 0.0288i)4-s + (0.495 − 0.194i)5-s + (−1.11 − 0.151i)6-s + (0.671 − 0.741i)7-s + (0.0432 + 0.999i)8-s + (−0.250 − 0.0773i)9-s + (−0.201 − 0.492i)10-s + (−0.348 − 1.12i)11-s + (−0.135 + 1.11i)12-s + (−1.29 − 0.294i)13-s + (−0.751 − 0.660i)14-s + (−0.133 − 0.583i)15-s + (0.998 − 0.0576i)16-s + (0.512 + 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0693727 - 1.38328i\)
\(L(\frac12)\) \(\approx\) \(0.0693727 - 1.38328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0204 + 1.41i)T \)
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

−10.82954480628089439117633010532, −10.17705019203844200373449123484, −9.126952441389624777934654750949, −7.895672284504080889580748641182, −7.56324096699346203870751585641, −5.88615658108922176214157645170, −4.91318342406124717444869199381, −3.43533677602527199562408431250, −2.08125358672085832643016129684, −0.984406043999087072706975755993, 2.47745022258447110716540379752, 4.30934046605671642607790540734, 4.90095168828932534120149606338, 5.82302550698623449501981552871, 7.16265782284890276227929908166, 7.976694883774756651946357740822, 9.202070309575813521449799100497, 9.761972167889024795072429305785, 10.29564761307470317777585422471, 11.86371357132259892877950973254

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