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Properties

Label 2-1792-16.5-c1-0-12
Degree $2$
Conductor $1792$
Sign $-0.991 + 0.130i$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
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  + (−1.18 + 1.18i)3-s + (1.87 + 1.87i)5-s + i·7-s + 0.202i·9-s + (0.584 + 0.584i)11-s + (−3.94 + 3.94i)13-s − 4.44·15-s + 1.74·17-s + (−4.19 + 4.19i)19-s + (−1.18 − 1.18i)21-s − 3.04i·23-s + 2.05i·25-s + (−3.78 − 3.78i)27-s + (4.43 − 4.43i)29-s − 7.90·31-s + ⋯
L(s)  = 1  + (−0.682 + 0.682i)3-s + (0.839 + 0.839i)5-s + 0.377i·7-s + 0.0675i·9-s + (0.176 + 0.176i)11-s + (−1.09 + 1.09i)13-s − 1.14·15-s + 0.424·17-s + (−0.962 + 0.962i)19-s + (−0.258 − 0.258i)21-s − 0.634i·23-s + 0.411i·25-s + (−0.728 − 0.728i)27-s + (0.823 − 0.823i)29-s − 1.42·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.991 + 0.130i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9969670237\)
\(L(\frac12)\) \(\approx\) \(0.9969670237\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - iT \)
good3 \( 1 + (1.18 - 1.18i)T - 3iT^{2} \)
5 \( 1 + (-1.87 - 1.87i)T + 5iT^{2} \)
11 \( 1 + (-0.584 - 0.584i)T + 11iT^{2} \)
13 \( 1 + (3.94 - 3.94i)T - 13iT^{2} \)
17 \( 1 - 1.74T + 17T^{2} \)
19 \( 1 + (4.19 - 4.19i)T - 19iT^{2} \)
23 \( 1 + 3.04iT - 23T^{2} \)
29 \( 1 + (-4.43 + 4.43i)T - 29iT^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 + (-5.87 - 5.87i)T + 37iT^{2} \)
41 \( 1 + 1.38iT - 41T^{2} \)
43 \( 1 + (-1.73 - 1.73i)T + 43iT^{2} \)
47 \( 1 - 1.80T + 47T^{2} \)
53 \( 1 + (-9.73 - 9.73i)T + 53iT^{2} \)
59 \( 1 + (4.74 + 4.74i)T + 59iT^{2} \)
61 \( 1 + (3.10 - 3.10i)T - 61iT^{2} \)
67 \( 1 + (4.81 - 4.81i)T - 67iT^{2} \)
71 \( 1 - 1.11iT - 71T^{2} \)
73 \( 1 + 11.2iT - 73T^{2} \)
79 \( 1 - 7.61T + 79T^{2} \)
83 \( 1 + (-11.1 + 11.1i)T - 83iT^{2} \)
89 \( 1 + 0.428iT - 89T^{2} \)
97 \( 1 + 19.2T + 97T^{2} \)
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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

−9.797157085669951946015500756083, −9.234064175925585578362135726524, −8.101850156745219065884889149574, −7.15896855662269970402346817278, −6.26283224081949694775750737609, −5.81505491573215130341227112324, −4.76056122587689513023292808407, −4.11662337283760799012351115442, −2.65722706390927916651561699264, −1.93823660917937162263220987702, 0.40215025887380247337507949872, 1.37769805761634180513217009660, 2.57801880632065969856862008727, 3.90749997354768036526409978050, 5.20963341770963638119803619174, 5.45108460820214708239709816461, 6.46909276821208803067257670921, 7.18523887696017753248870897305, 7.960737490483452430555117749919, 9.056636611716354048433551884698

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