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Properties

Label 2-1352-13.4-c1-0-22
Degree $2$
Conductor $1352$
Sign $0.890 - 0.454i$
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  + (−0.5 + 0.866i)3-s + i·5-s + (4.33 − 2.5i)7-s + (1 + 1.73i)9-s + (1.73 + i)11-s + (−0.866 − 0.5i)15-s + (−1.5 − 2.59i)17-s + (1.73 − i)19-s + 5i·21-s + (2 − 3.46i)23-s + 4·25-s − 5·27-s + (3 − 5.19i)29-s + 4i·31-s + (−1.73 + 0.999i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + 0.447i·5-s + (1.63 − 0.944i)7-s + (0.333 + 0.577i)9-s + (0.522 + 0.301i)11-s + (−0.223 − 0.129i)15-s + (−0.363 − 0.630i)17-s + (0.397 − 0.229i)19-s + 1.09i·21-s + (0.417 − 0.722i)23-s + 0.800·25-s − 0.962·27-s + (0.557 − 0.964i)29-s + 0.718i·31-s + (−0.301 + 0.174i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $0.890 - 0.454i$
Analytic conductor: \(10.7957\)
Root analytic conductor: \(3.28569\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (1161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :1/2),\ 0.890 - 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.987037038\)
\(L(\frac12)\) \(\approx\) \(1.987037038\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 + (-4.33 + 2.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.73 - i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.73 + i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (9.52 + 5.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.92 - 4i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9iT - 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (-5.19 + 3i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.19 - 3i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.06 - 3.5i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 + (-8.66 - 5i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.66 + 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.828396482441445330519962602808, −8.869595167360800162528981679554, −7.904633912967233763830519555072, −7.30807965185695685931448855228, −6.52494852169395697927909636057, −5.04189248480755930685929310993, −4.74713927219850947945654995439, −3.85725031879058091484370889271, −2.38939388308977133714152700041, −1.16326587446068106245435608355, 1.17454344268463747384849374258, 1.90922393461162340517652351645, 3.45704971422960311177900691371, 4.65142102538465163895346004213, 5.33756101457568575114146852483, 6.18719537761229473246899212685, 7.11584689957780865487201945311, 8.013229817688853375932979838742, 8.764269641057153908610105137540, 9.217208960672570198340339327360

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