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Properties

Label 2-1352-13.10-c1-0-26
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  + (−0.5 − 0.866i)3-s + 2i·5-s + (−0.866 − 0.5i)7-s + (1 − 1.73i)9-s + (−0.866 + 0.5i)11-s + (1.73 − i)15-s + (1.5 − 2.59i)17-s + (−6.06 − 3.5i)19-s + 0.999i·21-s + (0.5 + 0.866i)23-s + 25-s − 5·27-s + (−1.5 − 2.59i)29-s + 8i·31-s + (0.866 + 0.499i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 0.894i·5-s + (−0.327 − 0.188i)7-s + (0.333 − 0.577i)9-s + (−0.261 + 0.150i)11-s + (0.447 − 0.258i)15-s + (0.363 − 0.630i)17-s + (−1.39 − 0.802i)19-s + 0.218i·21-s + (0.104 + 0.180i)23-s + 0.200·25-s − 0.962·27-s + (−0.278 − 0.482i)29-s + 1.43i·31-s + (0.150 + 0.0870i)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

\(L(1)\) \(\approx\) \(1.000009179\)
\(L(\frac12)\) \(\approx\) \(1.000009179\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.06 + 3.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8iT - 31T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.52 + 5.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (7.79 + 4.5i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.59 - 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.33 - 2.5i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + (-0.866 + 0.5i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.866 - 0.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.416097328939513695203482190812, −8.594316257215351956099548203919, −7.38514480951161442775157118990, −6.94157465039014061603711463672, −6.32526931506874063884761838113, −5.29412424309469960973757172662, −4.10478047157939660301451371574, −3.14161163265236439477912978051, −2.07583734964672745384858425847, −0.43872865594652725763861571586, 1.38651376352792272666766607960, 2.72116057058191357534625294106, 4.18952440107379396271921924835, 4.57663600176189263592152630357, 5.73979978989239899152442985381, 6.22760060373349682421505218178, 7.70609517667195738974538299126, 8.143769620216233303830859104145, 9.204309651703589947399245459004, 9.720992541602266287580911783802

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