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Properties

Label 2-351-117.103-c1-0-3
Degree $2$
Conductor $351$
Sign $-0.503 - 0.864i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + (−1.97 + 1.14i)2-s + (1.60 − 2.78i)4-s + (2.78 + 1.60i)5-s + (−2.09 + 1.21i)7-s + 2.76i·8-s − 7.34·10-s + (−1.27 + 0.737i)11-s + (3.56 + 0.535i)13-s + (2.76 − 4.79i)14-s + (0.0535 + 0.0927i)16-s + 5.12·17-s − 1.13i·19-s + (8.94 − 5.16i)20-s + (1.68 − 2.91i)22-s + (−4.61 + 7.99i)23-s + ⋯
L(s)  = 1  + (−1.39 + 0.807i)2-s + (0.803 − 1.39i)4-s + (1.24 + 0.719i)5-s + (−0.793 + 0.457i)7-s + 0.978i·8-s − 2.32·10-s + (−0.385 + 0.222i)11-s + (0.988 + 0.148i)13-s + (0.739 − 1.28i)14-s + (0.0133 + 0.0231i)16-s + 1.24·17-s − 0.259i·19-s + (2.00 − 1.15i)20-s + (0.358 − 0.621i)22-s + (−0.962 + 1.66i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.503 - 0.864i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.503 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363709 + 0.632691i\)
\(L(\frac12)\) \(\approx\) \(0.363709 + 0.632691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-3.56 - 0.535i)T \)
good2 \( 1 + (1.97 - 1.14i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.78 - 1.60i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.09 - 1.21i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.27 - 0.737i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 5.12T + 17T^{2} \)
19 \( 1 + 1.13iT - 19T^{2} \)
23 \( 1 + (4.61 - 7.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.487 - 0.844i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.16 + 1.82i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.22iT - 37T^{2} \)
41 \( 1 + (3.47 + 2.00i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.33 - 7.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.33 - 0.769i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.739T + 53T^{2} \)
59 \( 1 + (-6.72 - 3.88i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.06 + 7.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.669 + 0.386i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.01iT - 71T^{2} \)
73 \( 1 - 9.21iT - 73T^{2} \)
79 \( 1 + (1.86 + 3.23i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.3 + 7.13i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.21iT - 89T^{2} \)
97 \( 1 + (-13.1 + 7.61i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.39605913725167664827953115343, −10.30812319084014397663293603774, −9.783058636224021405330919181395, −9.199793390943911545978974606857, −8.051275625108029955878096723557, −7.10621289258672487186625161166, −6.08978824328115749461942874428, −5.71598551939164270322311161339, −3.25849620739321154860680287387, −1.67947960739461193374193747883, 0.834585319759172797118891515126, 2.15834151408954920685017656590, 3.55176798966882733989552492210, 5.42909533449111057654873451498, 6.42013593561398686593450030987, 7.84219745391039143602851071097, 8.696694872495470225474086144079, 9.439567080330525278565517486274, 10.30248940176026343581302074660, 10.56858588380328656068902176140

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