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Properties

Label 2-349-349.348-c1-0-18
Degree $2$
Conductor $349$
Sign $0.695 + 0.718i$
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  − 3-s + 2·4-s + 2·5-s − 4.47i·7-s − 2·9-s − 4.47i·11-s − 2·12-s + 4.47i·13-s − 2·15-s + 4·16-s + 3·17-s − 5·19-s + 4·20-s + 4.47i·21-s + 23-s + ⋯
L(s)  = 1  − 0.577·3-s + 4-s + 0.894·5-s − 1.69i·7-s − 0.666·9-s − 1.34i·11-s − 0.577·12-s + 1.24i·13-s − 0.516·15-s + 16-s + 0.727·17-s − 1.14·19-s + 0.894·20-s + 0.975i·21-s + 0.208·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $0.695 + 0.718i$
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: \((2,\ 349,\ (\ :1/2),\ 0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37798 - 0.583544i\)
\(L(\frac12)\) \(\approx\) \(1.37798 - 0.583544i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (13 + 13.4i)T \)
good2 \( 1 - 2T^{2} \)
3 \( 1 + T + 3T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + 4.47iT - 7T^{2} \)
11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 - 4.47iT - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 4.47iT - 43T^{2} \)
47 \( 1 - 8.94iT - 47T^{2} \)
53 \( 1 - 4.47iT - 53T^{2} \)
59 \( 1 + 4.47iT - 59T^{2} \)
61 \( 1 - 8.94iT - 61T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 - 8.94iT - 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 - 4.47iT - 79T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 + 8.94iT - 89T^{2} \)
97 \( 1 - 13.4iT - 97T^{2} \)
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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

−11.12367727251626748001177459835, −10.73447402918863351155831343460, −9.839257282582249937589771904985, −8.502230174053779631645065150992, −7.36130926723663591819888226407, −6.32078027152616484061343821959, −5.93068955907541386639922858133, −4.31393725584197670499754263938, −2.88551686569692652085333624868, −1.20313764724857323949867033725, 2.04582351986007842577548054606, 2.83570918859238160672004619638, 5.14619659011423115373513767235, 5.82785080793056503886606525470, 6.44393701970230526130647023645, 7.82550652207175712021595156171, 8.852948300127122231453789771735, 10.00322431469670045263320342590, 10.62362697135270406663297660139, 11.86343872111650149906984348679

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