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Properties

Label 2-1352-13.9-c1-0-34
Degree $2$
Conductor $1352$
Sign $-0.309 + 0.950i$
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  + (1.40 − 2.42i)3-s + 2.80·5-s + (−0.900 − 1.56i)7-s + (−2.42 − 4.20i)9-s + (3.20 − 5.54i)11-s + (3.92 − 6.79i)15-s + (2.81 + 4.87i)17-s + (0.832 + 1.44i)19-s − 5.04·21-s + (−1.54 + 2.68i)23-s + 2.85·25-s − 5.18·27-s + (−3.03 + 5.26i)29-s − 3.58·31-s + (−8.97 − 15.5i)33-s + ⋯
L(s)  = 1  + (0.808 − 1.40i)3-s + 1.25·5-s + (−0.340 − 0.589i)7-s + (−0.808 − 1.40i)9-s + (0.965 − 1.67i)11-s + (1.01 − 1.75i)15-s + (0.682 + 1.18i)17-s + (0.190 + 0.330i)19-s − 1.10·21-s + (−0.322 + 0.559i)23-s + 0.570·25-s − 0.998·27-s + (−0.564 + 0.977i)29-s − 0.643·31-s + (−1.56 − 2.70i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.309 + 0.950i$
Analytic conductor: \(10.7957\)
Root analytic conductor: \(3.28569\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :1/2),\ -0.309 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.765718893\)
\(L(\frac12)\) \(\approx\) \(2.765718893\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 2.42i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.80T + 5T^{2} \)
7 \( 1 + (0.900 + 1.56i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.20 + 5.54i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.81 - 4.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.832 - 1.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.54 - 2.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.03 - 5.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 + (-2.57 + 4.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.54 - 6.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.42 - 2.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.51T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + (-4.63 - 8.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.41 - 11.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.08 + 1.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.45 + 9.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.95T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + 0.0392T + 83T^{2} \)
89 \( 1 + (-3.72 + 6.44i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.236 - 0.408i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.157953150632465108005028952682, −8.562906161125705256014678273988, −7.73983047769077486518032773258, −6.89994264256077944632226625407, −6.05195186517411047908882590831, −5.74360806195077152571564453305, −3.76417610862966893635938840117, −3.08970351664827653706125223152, −1.76784391301993954875748550214, −1.13312627759084788567904874891, 1.95299860516328204223758625380, 2.71698493606468383490978345665, 3.84151336093441117699054220746, 4.75284987886942064445396257172, 5.45163851254367039229619644410, 6.48361375462457409048061846551, 7.45079463718952600425028836396, 8.658174750699844245504040270682, 9.403485517323229611357068718706, 9.734949627464978980016856511592

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