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Properties

Label 2-349-349.348-c1-0-4
Degree $2$
Conductor $349$
Sign $-0.491 - 0.870i$
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  + 0.788i·2-s − 0.670·3-s + 1.37·4-s − 1.84·5-s − 0.529i·6-s + 0.0300i·7-s + 2.66i·8-s − 2.54·9-s − 1.45i·10-s + 3.59i·11-s − 0.924·12-s + 6.25i·13-s − 0.0236·14-s + 1.23·15-s + 0.654·16-s + 3.51·17-s + ⋯
L(s)  = 1  + 0.557i·2-s − 0.387·3-s + 0.688·4-s − 0.825·5-s − 0.216i·6-s + 0.0113i·7-s + 0.941i·8-s − 0.849·9-s − 0.460i·10-s + 1.08i·11-s − 0.266·12-s + 1.73i·13-s − 0.00632·14-s + 0.319·15-s + 0.163·16-s + 0.853·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $-0.491 - 0.870i$
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: \((2,\ 349,\ (\ :1/2),\ -0.491 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518344 + 0.887694i\)
\(L(\frac12)\) \(\approx\) \(0.518344 + 0.887694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (-9.18 - 16.2i)T \)
good2 \( 1 - 0.788iT - 2T^{2} \)
3 \( 1 + 0.670T + 3T^{2} \)
5 \( 1 + 1.84T + 5T^{2} \)
7 \( 1 - 0.0300iT - 7T^{2} \)
11 \( 1 - 3.59iT - 11T^{2} \)
13 \( 1 - 6.25iT - 13T^{2} \)
17 \( 1 - 3.51T + 17T^{2} \)
19 \( 1 - 3.38T + 19T^{2} \)
23 \( 1 + 1.78T + 23T^{2} \)
29 \( 1 + 4.91T + 29T^{2} \)
31 \( 1 + 3.21T + 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 + 8.41T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 - 2.19iT - 47T^{2} \)
53 \( 1 + 12.0iT - 53T^{2} \)
59 \( 1 - 9.14iT - 59T^{2} \)
61 \( 1 + 0.831iT - 61T^{2} \)
67 \( 1 + 3.48T + 67T^{2} \)
71 \( 1 - 7.63iT - 71T^{2} \)
73 \( 1 - 8.96T + 73T^{2} \)
79 \( 1 + 9.94iT - 79T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 + 8.20iT - 89T^{2} \)
97 \( 1 + 6.03iT - 97T^{2} \)
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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

−11.74563505845736907058594920994, −11.24456888425685165379139082215, −9.972567389197473104529315155559, −8.863419593947714412185541696070, −7.70543411660818064624975836033, −7.13894324574208023281439686890, −6.09770528338621391136590100926, −5.06598645411663483434711542447, −3.73901405076749634552927442395, −2.09084185605050909738753458956, 0.73463520483830942062536728043, 2.92557919311175919553316255968, 3.58179160828279704716543950514, 5.45987550162015353689385479386, 6.12113511166132415348888682194, 7.66241830799089522576093862254, 8.082640986831178293489887448731, 9.575065299217021904935651815215, 10.66298751344647194929796137325, 11.22151148181365783042344319044

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