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Properties

Label 2-349-349.348-c1-0-27
Degree $2$
Conductor $349$
Sign $0.995 + 0.0897i$
Analytic cond. $2.78677$
Root an. cond. $1.66936$
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  − 2.61i·2-s − 2.83·3-s − 4.86·4-s − 3.82·5-s + 7.43i·6-s − 2.86i·7-s + 7.49i·8-s + 5.05·9-s + 10.0i·10-s − 5.86i·11-s + 13.7·12-s + 1.62i·13-s − 7.50·14-s + 10.8·15-s + 9.90·16-s − 1.17·17-s + ⋯
L(s)  = 1  − 1.85i·2-s − 1.63·3-s − 2.43·4-s − 1.71·5-s + 3.03i·6-s − 1.08i·7-s + 2.64i·8-s + 1.68·9-s + 3.16i·10-s − 1.76i·11-s + 3.98·12-s + 0.449i·13-s − 2.00·14-s + 2.80·15-s + 2.47·16-s − 0.284·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $0.995 + 0.0897i$
Analytic conductor: \(2.78677\)
Root analytic conductor: \(1.66936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 349,\ (\ :1/2),\ 0.995 + 0.0897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0363829 - 0.00163546i\)
\(L(\frac12)\) \(\approx\) \(0.0363829 - 0.00163546i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (18.6 + 1.67i)T \)
good2 \( 1 + 2.61iT - 2T^{2} \)
3 \( 1 + 2.83T + 3T^{2} \)
5 \( 1 + 3.82T + 5T^{2} \)
7 \( 1 + 2.86iT - 7T^{2} \)
11 \( 1 + 5.86iT - 11T^{2} \)
13 \( 1 - 1.62iT - 13T^{2} \)
17 \( 1 + 1.17T + 17T^{2} \)
19 \( 1 + 3.55T + 19T^{2} \)
23 \( 1 - 4.12T + 23T^{2} \)
29 \( 1 + 7.29T + 29T^{2} \)
31 \( 1 - 1.51T + 31T^{2} \)
37 \( 1 - 4.67T + 37T^{2} \)
41 \( 1 + 8.12T + 41T^{2} \)
43 \( 1 - 6.29iT - 43T^{2} \)
47 \( 1 + 0.679iT - 47T^{2} \)
53 \( 1 + 6.90iT - 53T^{2} \)
59 \( 1 + 2.76iT - 59T^{2} \)
61 \( 1 + 2.47iT - 61T^{2} \)
67 \( 1 + 2.33T + 67T^{2} \)
71 \( 1 - 1.45iT - 71T^{2} \)
73 \( 1 + 1.17T + 73T^{2} \)
79 \( 1 + 4.86iT - 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 - 12.6iT - 97T^{2} \)
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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

−10.99600905180637912824225770367, −10.39866891962182521960762006862, −8.952574172341717110903688901706, −7.889419691173755083601321566394, −6.56660691667240961847120786197, −5.02462101687551094960806582451, −4.14067369622309908881513760644, −3.44734889256409054160137042120, −0.915802655055591902917174228656, −0.04501360637592440633355234031, 4.20274948711371571764383631675, 4.86698682013819637711644460758, 5.70424204061404315083640467221, 6.83507574841861276639582782614, 7.31900727998433087047173119320, 8.313501476772638139244258254120, 9.353548810943548170739366527367, 10.65216110174526375802499465243, 11.75873098765698342774024818413, 12.41581552766268363951150398812

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