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Properties

Label 2-1352-13.10-c1-0-10
Degree $2$
Conductor $1352$
Sign $0.669 + 0.743i$
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  + (−1.66 − 2.88i)3-s − 0.932i·5-s + (2.99 + 1.72i)7-s + (−4.04 + 7.01i)9-s + (1.60 − 0.929i)11-s + (−2.69 + 1.55i)15-s + (0.109 − 0.189i)17-s + (4.42 + 2.55i)19-s − 11.5i·21-s + (3.74 + 6.49i)23-s + 4.13·25-s + 16.9·27-s + (1.59 + 2.76i)29-s + 7.91i·31-s + (−5.36 − 3.09i)33-s + ⋯
L(s)  = 1  + (−0.961 − 1.66i)3-s − 0.417i·5-s + (1.13 + 0.653i)7-s + (−1.34 + 2.33i)9-s + (0.485 − 0.280i)11-s + (−0.694 + 0.401i)15-s + (0.0264 − 0.0458i)17-s + (1.01 + 0.585i)19-s − 2.51i·21-s + (0.781 + 1.35i)23-s + 0.826·25-s + 3.26·27-s + (0.296 + 0.514i)29-s + 1.42i·31-s + (−0.933 − 0.538i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $0.669 + 0.743i$
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: \((2,\ 1352,\ (\ :1/2),\ 0.669 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.389604651\)
\(L(\frac12)\) \(\approx\) \(1.389604651\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 2.88i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.932iT - 5T^{2} \)
7 \( 1 + (-2.99 - 1.72i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.60 + 0.929i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.109 + 0.189i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.42 - 2.55i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.74 - 6.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.59 - 2.76i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.91iT - 31T^{2} \)
37 \( 1 + (-3.69 + 2.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.83 - 1.63i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.29 - 3.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.30iT - 47T^{2} \)
53 \( 1 + 2.68T + 53T^{2} \)
59 \( 1 + (7.99 + 4.61i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.11 + 1.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.84 + 2.21i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.25 - 1.30i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.28iT - 73T^{2} \)
79 \( 1 - 7.33T + 79T^{2} \)
83 \( 1 + 3.58iT - 83T^{2} \)
89 \( 1 + (4.39 - 2.53i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.6 + 6.70i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.259481505060567246914013594657, −8.418081270660541469400817740538, −7.81319728538261809351249968510, −7.04823414235215559882622485874, −6.22889167279628207908536342015, −5.24046890812843229190323480607, −5.02977937331137778467156186654, −3.07400088860853932384665965297, −1.65937118162924226499083032710, −1.15798964184787289497155380671, 0.832952083068983214088964823855, 2.86304191695279697190396938223, 4.02424629106876006783434189418, 4.61718585782602101297092627100, 5.24378141219322273150543755222, 6.29523324313466538962071296501, 7.09407474943512485550492991642, 8.260600885058644773167696846376, 9.183391896855279559513855291332, 9.853917757851888205116634835165

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