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Properties

Label 2-351-13.12-c1-0-13
Degree $2$
Conductor $351$
Sign $-0.916 + 0.401i$
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  − 2.07i·2-s − 2.30·4-s + 2.70i·5-s − 4.77i·7-s + 0.628i·8-s + 5.60·10-s − 2.07i·11-s + (3.30 − 1.44i)13-s − 9.90·14-s − 3.30·16-s − 6.90·17-s − 3.33i·19-s − 6.22i·20-s − 4.30·22-s + 3.90·23-s + ⋯
L(s)  = 1  − 1.46i·2-s − 1.15·4-s + 1.20i·5-s − 1.80i·7-s + 0.222i·8-s + 1.77·10-s − 0.625i·11-s + (0.916 − 0.401i)13-s − 2.64·14-s − 0.825·16-s − 1.67·17-s − 0.764i·19-s − 1.39i·20-s − 0.917·22-s + 0.814·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.916 + 0.401i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.916 + 0.401i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.247523 - 1.18234i\)
\(L(\frac12)\) \(\approx\) \(0.247523 - 1.18234i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.30 + 1.44i)T \)
good2 \( 1 + 2.07iT - 2T^{2} \)
5 \( 1 - 2.70iT - 5T^{2} \)
7 \( 1 + 4.77iT - 7T^{2} \)
11 \( 1 + 2.07iT - 11T^{2} \)
17 \( 1 + 6.90T + 17T^{2} \)
19 \( 1 + 3.33iT - 19T^{2} \)
23 \( 1 - 3.90T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 2.89iT - 31T^{2} \)
37 \( 1 - 7.66iT - 37T^{2} \)
41 \( 1 + 0.628iT - 41T^{2} \)
43 \( 1 - 5.90T + 43T^{2} \)
47 \( 1 - 5.59iT - 47T^{2} \)
53 \( 1 - 6.90T + 53T^{2} \)
59 \( 1 + 7.28iT - 59T^{2} \)
61 \( 1 - 9.60T + 61T^{2} \)
67 \( 1 - 8.10iT - 67T^{2} \)
71 \( 1 + 2.51iT - 71T^{2} \)
73 \( 1 - 4.33iT - 73T^{2} \)
79 \( 1 - 2.69T + 79T^{2} \)
83 \( 1 - 2.26iT - 83T^{2} \)
89 \( 1 + 2.51iT - 89T^{2} \)
97 \( 1 - 9.55iT - 97T^{2} \)
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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

−10.89791490949437804092416529320, −10.70335000059984551599574943878, −9.707386149810652154667244475779, −8.500546002231209255156324874746, −7.07988877715287716755189134412, −6.53199196396501089663942454079, −4.47248180304230403674824128249, −3.59446640404219791966699780478, −2.67088886997655001317202829921, −0.875026792784839882645955175535, 2.13511136298671368879275543425, 4.40012917987304812840857933909, 5.28344508871044443366439117756, 6.02099139000528210825966220991, 6.99668786996931775435201909478, 8.425587997907646382552450461366, 8.746344894705470805512633701044, 9.341264439642899578820589406581, 11.10957216614997461673255408434, 12.12903204219803073769515112190

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