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Properties

Label 2-1408-176.131-c1-0-37
Degree $2$
Conductor $1408$
Sign $-0.268 + 0.963i$
Analytic cond. $11.2429$
Root an. cond. $3.35304$
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  + (−1.15 + 1.15i)3-s + (2.51 − 2.51i)5-s − 4.37i·7-s + 0.313i·9-s + (0.130 − 3.31i)11-s + (2.19 − 2.19i)13-s + 5.82i·15-s + 0.470i·17-s + (0.632 + 0.632i)19-s + (5.07 + 5.07i)21-s − 6.62·23-s − 7.61i·25-s + (−3.84 − 3.84i)27-s + (0.0549 − 0.0549i)29-s + 0.184i·31-s + ⋯
L(s)  = 1  + (−0.669 + 0.669i)3-s + (1.12 − 1.12i)5-s − 1.65i·7-s + 0.104i·9-s + (0.0393 − 0.999i)11-s + (0.609 − 0.609i)13-s + 1.50i·15-s + 0.114i·17-s + (0.145 + 0.145i)19-s + (1.10 + 1.10i)21-s − 1.38·23-s − 1.52i·25-s + (−0.739 − 0.739i)27-s + (0.0102 − 0.0102i)29-s + 0.0331i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $-0.268 + 0.963i$
Analytic conductor: \(11.2429\)
Root analytic conductor: \(3.35304\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1408} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1408,\ (\ :1/2),\ -0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.348831357\)
\(L(\frac12)\) \(\approx\) \(1.348831357\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-0.130 + 3.31i)T \)
good3 \( 1 + (1.15 - 1.15i)T - 3iT^{2} \)
5 \( 1 + (-2.51 + 2.51i)T - 5iT^{2} \)
7 \( 1 + 4.37iT - 7T^{2} \)
13 \( 1 + (-2.19 + 2.19i)T - 13iT^{2} \)
17 \( 1 - 0.470iT - 17T^{2} \)
19 \( 1 + (-0.632 - 0.632i)T + 19iT^{2} \)
23 \( 1 + 6.62T + 23T^{2} \)
29 \( 1 + (-0.0549 + 0.0549i)T - 29iT^{2} \)
31 \( 1 - 0.184iT - 31T^{2} \)
37 \( 1 + (3.55 - 3.55i)T - 37iT^{2} \)
41 \( 1 + 3.77T + 41T^{2} \)
43 \( 1 + (4.82 - 4.82i)T - 43iT^{2} \)
47 \( 1 + 0.258iT - 47T^{2} \)
53 \( 1 + (5.27 - 5.27i)T - 53iT^{2} \)
59 \( 1 + (-1.64 - 1.64i)T + 59iT^{2} \)
61 \( 1 + (-9.05 + 9.05i)T - 61iT^{2} \)
67 \( 1 + (1.83 - 1.83i)T - 67iT^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + 1.80T + 79T^{2} \)
83 \( 1 + (-8.61 - 8.61i)T + 83iT^{2} \)
89 \( 1 + 14.8iT - 89T^{2} \)
97 \( 1 + 9.13T + 97T^{2} \)
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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

−9.707730650025826035777607040796, −8.430119882828957841936565611577, −7.974239373730890798252165912933, −6.59792318933192961854619662967, −5.83143965247881821377796837649, −5.16854645999587571413010563695, −4.34654916667758368660224442440, −3.48241084585034040535193002928, −1.65894519594158264891374899691, −0.58388797809806964607179589520, 1.84991561131201167153138485781, 2.27769215673180404592955383114, 3.59620963987555883407375593563, 5.21150348615854660729703833132, 5.81833468891456287125749752392, 6.54798223082935785394236378437, 6.90537031553292001399271849302, 8.167604647104151299782810012883, 9.226772352748811134593859578638, 9.685536009708288283425112013320

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