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Properties

Label 2-1456-13.10-c1-0-34
Degree $2$
Conductor $1456$
Sign $0.914 + 0.405i$
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  + (1.14 + 1.98i)3-s − 0.901i·5-s + (−0.866 − 0.5i)7-s + (−1.12 + 1.94i)9-s + (3.75 − 2.16i)11-s + (−0.426 − 3.58i)13-s + (1.78 − 1.03i)15-s + (2.53 − 4.38i)17-s + (−5.34 − 3.08i)19-s − 2.29i·21-s + (−4.22 − 7.31i)23-s + 4.18·25-s + 1.72·27-s + (1.09 + 1.89i)29-s + 0.873i·31-s + ⋯
L(s)  = 1  + (0.661 + 1.14i)3-s − 0.403i·5-s + (−0.327 − 0.188i)7-s + (−0.374 + 0.648i)9-s + (1.13 − 0.653i)11-s + (−0.118 − 0.992i)13-s + (0.461 − 0.266i)15-s + (0.614 − 1.06i)17-s + (−1.22 − 0.708i)19-s − 0.499i·21-s + (−0.881 − 1.52i)23-s + 0.837·25-s + 0.331·27-s + (0.203 + 0.352i)29-s + 0.156i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.914 + 0.405i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.914 + 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.997158309\)
\(L(\frac12)\) \(\approx\) \(1.997158309\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.426 + 3.58i)T \)
good3 \( 1 + (-1.14 - 1.98i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.901iT - 5T^{2} \)
11 \( 1 + (-3.75 + 2.16i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.53 + 4.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.34 + 3.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.22 + 7.31i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.09 - 1.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.873iT - 31T^{2} \)
37 \( 1 + (-0.124 + 0.0721i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.46 - 1.99i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.85 - 6.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.92iT - 47T^{2} \)
53 \( 1 - 1.69T + 53T^{2} \)
59 \( 1 + (-7.40 - 4.27i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.16 - 7.21i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.99 + 5.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.83 - 1.63i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.539iT - 73T^{2} \)
79 \( 1 + 6.53T + 79T^{2} \)
83 \( 1 + 13.2iT - 83T^{2} \)
89 \( 1 + (6.74 - 3.89i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.1 - 5.85i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.431237365851785634242008264766, −8.675206141722319357691792878503, −8.324291635553039776477930299283, −6.97714296230673591991963380834, −6.18960614404866909497093252588, −4.99923650867886967730503237381, −4.34789520204296367947524580897, −3.43136582754164256866401951049, −2.67006789436865966804824840152, −0.76263576618644116872168061932, 1.58198973376296812972664879175, 2.09746944184233241940880548235, 3.47799420301920960327571292515, 4.21808227258054709539142257010, 5.74604131047364579455792249167, 6.67100315335234441025146330223, 6.95210414821615145336176365619, 8.006623299817004049347142256283, 8.578236484820953208375640810006, 9.509100334348379596446727402002

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