Properties

Label 2-392-392.109-c1-0-17
Degree $2$
Conductor $392$
Sign $0.175 - 0.984i$
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  + (1.21 + 0.726i)2-s + (0.0230 + 0.152i)3-s + (0.943 + 1.76i)4-s + (0.525 + 0.206i)5-s + (−0.0831 + 0.202i)6-s + (−2.63 − 0.225i)7-s + (−0.137 + 2.82i)8-s + (2.84 − 0.877i)9-s + (0.487 + 0.632i)10-s + (−1.80 + 5.84i)11-s + (−0.247 + 0.184i)12-s + (6.52 − 1.48i)13-s + (−3.03 − 2.19i)14-s + (−0.0194 + 0.0850i)15-s + (−2.22 + 3.32i)16-s + (−4.09 + 2.79i)17-s + ⋯
L(s)  = 1  + (0.857 + 0.514i)2-s + (0.0132 + 0.0882i)3-s + (0.471 + 0.881i)4-s + (0.235 + 0.0922i)5-s + (−0.0339 + 0.0824i)6-s + (−0.996 − 0.0853i)7-s + (−0.0487 + 0.998i)8-s + (0.947 − 0.292i)9-s + (0.154 + 0.199i)10-s + (−0.543 + 1.76i)11-s + (−0.0715 + 0.0533i)12-s + (1.81 − 0.413i)13-s + (−0.810 − 0.585i)14-s + (−0.00501 + 0.0219i)15-s + (−0.555 + 0.831i)16-s + (−0.994 + 0.677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.175 - 0.984i$
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: \((2,\ 392,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69175 + 1.41616i\)
\(L(\frac12)\) \(\approx\) \(1.69175 + 1.41616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 - 0.726i)T \)
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

−11.80323325752945707957112783970, −10.49086568703608752384985835774, −9.845404052693547331049553009120, −8.618951744715989482053238295900, −7.46016313797398188808772464614, −6.63633376833077624986166898850, −5.91351212838586364064154793429, −4.47389115707823614493517026570, −3.74287173320420176904447037374, −2.23301984907500148864866645072, 1.31058350834678202796772557059, 3.06188262765446432693808644004, 3.83041694749468737787056623323, 5.30365848012487677851507901524, 6.16545515534586811297479446244, 6.94556971630423415445528074793, 8.498901782080725238643137004564, 9.455610946497159592813341363718, 10.48126785179814215785295957698, 11.09847819783398347759751192113

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