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Properties

Label 2-1352-13.10-c1-0-12
Degree $2$
Conductor $1352$
Sign $0.252 - 0.967i$
Analytic cond. $10.7957$
Root an. cond. $3.28569$
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  + (0.780 + 1.35i)3-s + 0.561i·5-s + (1.35 + 0.780i)7-s + (0.280 − 0.486i)9-s + (1.35 − 0.780i)11-s + (−0.759 + 0.438i)15-s + (−2.5 + 4.33i)17-s + (−1.35 − 0.780i)19-s + 2.43i·21-s + (4.34 + 7.52i)23-s + 4.68·25-s + 5.56·27-s + (2.5 + 4.33i)29-s − 8i·31-s + (2.11 + 1.21i)33-s + ⋯
L(s)  = 1  + (0.450 + 0.780i)3-s + 0.251i·5-s + (0.511 + 0.295i)7-s + (0.0935 − 0.162i)9-s + (0.407 − 0.235i)11-s + (−0.196 + 0.113i)15-s + (−0.606 + 1.05i)17-s + (−0.310 − 0.179i)19-s + 0.532i·21-s + (0.905 + 1.56i)23-s + 0.936·25-s + 1.07·27-s + (0.464 + 0.804i)29-s − 1.43i·31-s + (0.367 + 0.212i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(10.7957\)
Root analytic conductor: \(3.28569\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :1/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.136573482\)
\(L(\frac12)\) \(\approx\) \(2.136573482\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.780 - 1.35i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.561iT - 5T^{2} \)
7 \( 1 + (-1.35 - 0.780i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.35 + 0.780i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.35 + 0.780i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.34 - 7.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8iT - 31T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.27 - 3.62i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.21 - 2.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 8.56T + 53T^{2} \)
59 \( 1 + (-1.35 - 0.780i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.62 - 8.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.7 - 6.78i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.05 - 2.34i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 14.2iT - 83T^{2} \)
89 \( 1 + (2.32 - 1.34i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.4 - 8.90i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.760309176451793264730774480212, −8.813330566236378269153947494821, −8.563503673218445272025572772137, −7.32170537615462062966045926435, −6.52687074114947220345517464008, −5.51100650726264300074622454072, −4.53677240379484448124956732544, −3.74582452348395738655745793709, −2.84008880530730141856658389543, −1.45308493977658499567647898229, 0.934972108495174345304872610848, 2.06958132103718470803977403720, 3.04084058216161408047456724508, 4.55760280706305624148894438581, 4.93585149407208575771518435350, 6.51364448393575040452276968352, 6.96250203968606348289574030503, 7.82663153784431282452434157155, 8.612458578833541203273907487852, 9.133377166695950927052875780267

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