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Properties

Label 2-351-13.12-c1-0-6
Degree $2$
Conductor $351$
Sign $0.0839 - 0.996i$
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  + 0.834i·2-s + 1.30·4-s + 1.92i·5-s − 1.08i·7-s + 2.75i·8-s − 1.60·10-s + 0.834i·11-s + (−0.302 + 3.59i)13-s + 0.908·14-s + 0.302·16-s + 3.90·17-s − 4.68i·19-s + 2.50i·20-s − 0.697·22-s − 6.90·23-s + ⋯
L(s)  = 1  + 0.590i·2-s + 0.651·4-s + 0.859i·5-s − 0.411i·7-s + 0.975i·8-s − 0.507·10-s + 0.251i·11-s + (−0.0839 + 0.996i)13-s + 0.242·14-s + 0.0756·16-s + 0.947·17-s − 1.07i·19-s + 0.560i·20-s − 0.148·22-s − 1.44·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0839 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0839 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.0839 - 0.996i$
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: \((2,\ 351,\ (\ :1/2),\ 0.0839 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16062 + 1.06692i\)
\(L(\frac12)\) \(\approx\) \(1.16062 + 1.06692i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.302 - 3.59i)T \)
good2 \( 1 - 0.834iT - 2T^{2} \)
5 \( 1 - 1.92iT - 5T^{2} \)
7 \( 1 + 1.08iT - 7T^{2} \)
11 \( 1 - 0.834iT - 11T^{2} \)
17 \( 1 - 3.90T + 17T^{2} \)
19 \( 1 + 4.68iT - 19T^{2} \)
23 \( 1 + 6.90T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 7.18iT - 31T^{2} \)
37 \( 1 + 6.09iT - 37T^{2} \)
41 \( 1 + 2.75iT - 41T^{2} \)
43 \( 1 + 4.90T + 43T^{2} \)
47 \( 1 + 5.26iT - 47T^{2} \)
53 \( 1 + 3.90T + 53T^{2} \)
59 \( 1 + 12.1iT - 59T^{2} \)
61 \( 1 - 2.39T + 61T^{2} \)
67 \( 1 - 5.76iT - 67T^{2} \)
71 \( 1 + 11.0iT - 71T^{2} \)
73 \( 1 + 10.7iT - 73T^{2} \)
79 \( 1 - 6.30T + 79T^{2} \)
83 \( 1 + 9.94iT - 83T^{2} \)
89 \( 1 + 11.0iT - 89T^{2} \)
97 \( 1 - 2.17iT - 97T^{2} \)
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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

−11.63384647152685889407677685548, −10.76124793049839915091860522652, −10.04624431999768610693338508783, −8.754111686848051270218513557382, −7.58638020979209194025753434969, −6.94363779676997666335468722464, −6.22033452059980029649527926257, −4.91309372780614003540004579981, −3.40178851514602004045499738586, −2.07998894306368308544518882492, 1.21972994149307260589189796226, 2.71459621845423622788610982830, 3.93818293060776003964499085978, 5.44076740166008921728038234956, 6.22528344089999355015508397554, 7.73330219730166271943306838566, 8.364132440250491637356021031561, 9.758387350549958605480792720823, 10.25095129729014016932868594002, 11.42352198109232754615445865708

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