Properties

Label 2-37e2-37.36-c1-0-59
Degree $2$
Conductor $1369$
Sign $0.164 + 0.986i$
Analytic cond. $10.9315$
Root an. cond. $3.30628$
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  i·2-s + 4-s + i·5-s + 2·7-s − 3i·8-s − 3·9-s + 10-s + 2·11-s − 2i·13-s − 2i·14-s − 16-s + 3i·17-s + 3i·18-s − 6i·19-s + i·20-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.5·4-s + 0.447i·5-s + 0.755·7-s − 1.06i·8-s − 9-s + 0.316·10-s + 0.603·11-s − 0.554i·13-s − 0.534i·14-s − 0.250·16-s + 0.727i·17-s + 0.707i·18-s − 1.37i·19-s + 0.223i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1369\)    =    \(37^{2}\)
Sign: $0.164 + 0.986i$
Analytic conductor: \(10.9315\)
Root analytic conductor: \(3.30628\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1369} (1368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1369,\ (\ :1/2),\ 0.164 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.084567210\)
\(L(\frac12)\) \(\approx\) \(2.084567210\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
good2 \( 1 + iT - 2T^{2} \)
3 \( 1 + 3T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + iT - 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 7iT - 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−9.518574035798762731419040068585, −8.627704879605355223176150547974, −7.895851122014243198290871213314, −6.82163947242747008542036682539, −6.26613157145651842087006719510, −5.15338786753145306060573166697, −4.05792798512623883656923640282, −2.96286592349509870531365920060, −2.32305371418684077330378160563, −0.918897384453553686228378030356, 1.40913665351903306798354418610, 2.54032857992879832688774485749, 3.83322059156717427375609800087, 5.04007458322309951563925848145, 5.64758926230507540537518691371, 6.46131305779098999461042783181, 7.41927212550470090055314017544, 8.037094102097097496330275314050, 8.817704877407745737109133813974, 9.469405250858333139856631744935

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