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Properties

Label 2-1456-7.4-c1-0-14
Degree $2$
Conductor $1456$
Sign $-0.839 - 0.543i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + (−1.31 + 2.27i)3-s + (1.45 + 2.51i)5-s + (1.29 − 2.30i)7-s + (−1.95 − 3.37i)9-s + (1.01 − 1.76i)11-s + 13-s − 7.62·15-s + (−1.99 + 3.46i)17-s + (3.48 + 6.02i)19-s + (3.54 + 5.97i)21-s + (−0.313 − 0.543i)23-s + (−1.71 + 2.96i)25-s + 2.37·27-s + 1.09·29-s + (−5.21 + 9.03i)31-s + ⋯
L(s)  = 1  + (−0.758 + 1.31i)3-s + (0.649 + 1.12i)5-s + (0.489 − 0.871i)7-s + (−0.650 − 1.12i)9-s + (0.307 − 0.531i)11-s + 0.277·13-s − 1.96·15-s + (−0.484 + 0.839i)17-s + (0.798 + 1.38i)19-s + (0.774 + 1.30i)21-s + (−0.0653 − 0.113i)23-s + (−0.342 + 0.593i)25-s + 0.456·27-s + 0.203·29-s + (−0.936 + 1.62i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.372262658\)
\(L(\frac12)\) \(\approx\) \(1.372262658\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 2.30i)T \)
13 \( 1 - T \)
good3 \( 1 + (1.31 - 2.27i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.45 - 2.51i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.01 + 1.76i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.99 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.48 - 6.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.313 + 0.543i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.09T + 29T^{2} \)
31 \( 1 + (5.21 - 9.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.54 - 2.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.521T + 41T^{2} \)
43 \( 1 + 0.329T + 43T^{2} \)
47 \( 1 + (-5.27 - 9.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.55 - 6.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.01 + 1.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.20 + 2.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.34 + 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.60T + 71T^{2} \)
73 \( 1 + (1.48 - 2.57i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.38 + 7.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + (-1.34 - 2.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.32T + 97T^{2} \)
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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

−10.11464261612822860417543499096, −9.362449237236165191745559238471, −8.321813974001086121081969525725, −7.29473513059209602695698808553, −6.31644965497671160653444741024, −5.80152452275423028983552996424, −4.79827776774988328201426924421, −3.87264386472837308465182091588, −3.21620169367916404783043569564, −1.52230311439115592947114837423, 0.63413296960133353482163636808, 1.69517365214444754705396716763, 2.48485630530719339821408337488, 4.42671710513991619948649022781, 5.38757108668009628105056261734, 5.67613259368450299143258179565, 6.80317569393867321405091828835, 7.39546977042108543642126021471, 8.424784109307848590118832844967, 9.139254443404621138571929281876

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