Properties

Label 2-392-392.109-c1-0-37
Degree $2$
Conductor $392$
Sign $0.383 + 0.923i$
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  + (0.804 − 1.16i)2-s + (0.111 + 0.739i)3-s + (−0.706 − 1.87i)4-s + (2.34 + 0.919i)5-s + (0.949 + 0.465i)6-s + (−2.33 − 1.23i)7-s + (−2.74 − 0.683i)8-s + (2.33 − 0.719i)9-s + (2.95 − 1.98i)10-s + (0.561 − 1.82i)11-s + (1.30 − 0.730i)12-s + (3.11 − 0.711i)13-s + (−3.32 + 1.72i)14-s + (−0.418 + 1.83i)15-s + (−3.00 + 2.64i)16-s + (2.96 − 2.01i)17-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)2-s + (0.0643 + 0.426i)3-s + (−0.353 − 0.935i)4-s + (1.04 + 0.411i)5-s + (0.387 + 0.189i)6-s + (−0.883 − 0.468i)7-s + (−0.970 − 0.241i)8-s + (0.777 − 0.239i)9-s + (0.934 − 0.627i)10-s + (0.169 − 0.549i)11-s + (0.376 − 0.210i)12-s + (0.864 − 0.197i)13-s + (−0.887 + 0.460i)14-s + (−0.108 + 0.473i)15-s + (−0.750 + 0.660i)16-s + (0.718 − 0.489i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.383 + 0.923i$
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: \((2,\ 392,\ (\ :1/2),\ 0.383 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68919 - 1.12734i\)
\(L(\frac12)\) \(\approx\) \(1.68919 - 1.12734i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.804 + 1.16i)T \)
7 \( 1 + (2.33 + 1.23i)T \)
good3 \( 1 + (-0.111 - 0.739i)T + (-2.86 + 0.884i)T^{2} \)
5 \( 1 + (-2.34 - 0.919i)T + (3.66 + 3.40i)T^{2} \)
11 \( 1 + (-0.561 + 1.82i)T + (-9.08 - 6.19i)T^{2} \)
13 \( 1 + (-3.11 + 0.711i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-2.96 + 2.01i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (0.698 - 0.403i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.36 - 1.61i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (3.29 - 6.84i)T + (-18.0 - 22.6i)T^{2} \)
31 \( 1 + (-1.00 + 1.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (10.4 - 0.782i)T + (36.5 - 5.51i)T^{2} \)
41 \( 1 + (-1.38 - 1.74i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (5.31 + 4.24i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (2.73 - 2.53i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (-5.65 - 0.424i)T + (52.4 + 7.89i)T^{2} \)
59 \( 1 + (2.46 - 0.968i)T + (43.2 - 40.1i)T^{2} \)
61 \( 1 + (14.4 - 1.08i)T + (60.3 - 9.09i)T^{2} \)
67 \( 1 + (-6.60 - 3.81i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (12.1 - 5.83i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (1.38 + 1.28i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (-7.02 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.2 - 2.79i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (5.11 - 1.57i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−10.84697557710739466462535443247, −10.33890174172291474889749227977, −9.630485434490508412152312787146, −8.910186430739771857421099672121, −7.01517151920536424375115982414, −6.16144041914913955246874428707, −5.23435257369103893839625559345, −3.78595215064290748617034608499, −3.10037458412444444905024656476, −1.38456720904869515479936842636, 1.92222209623285002372975511996, 3.51638810937995990042542011987, 4.83363250602431527334662809502, 5.93842208304347060251395775169, 6.51655053049647542204318509712, 7.53331571443729949062157121688, 8.685807295494559447047700635296, 9.453888697764303346692951678562, 10.33395519045264737162195681999, 11.97714103123064710012594780266

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