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Properties

Label 2-403-403.256-c1-0-35
Degree $2$
Conductor $403$
Sign $-0.0649 - 0.997i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.255 − 0.283i)2-s + (−2.18 − 0.464i)3-s + (0.193 − 1.84i)4-s − 3·5-s + (0.427 + 0.739i)6-s + (0.313 − 2.98i)7-s + (−1.19 + 0.865i)8-s + (1.82 + 0.813i)9-s + (0.766 + 0.851i)10-s + (−2.04 + 0.909i)11-s + (−1.28 + 3.94i)12-s + (3.24 + 1.57i)13-s + (−0.927 + 0.673i)14-s + (6.56 + 1.39i)15-s + (−3.07 − 0.654i)16-s + (−3.30 − 1.47i)17-s + ⋯
L(s)  = 1  + (−0.180 − 0.200i)2-s + (−1.26 − 0.268i)3-s + (0.0969 − 0.921i)4-s − 1.34·5-s + (0.174 + 0.301i)6-s + (0.118 − 1.12i)7-s + (−0.421 + 0.305i)8-s + (0.609 + 0.271i)9-s + (0.242 + 0.269i)10-s + (−0.615 + 0.274i)11-s + (−0.369 + 1.13i)12-s + (0.899 + 0.436i)13-s + (−0.247 + 0.180i)14-s + (1.69 + 0.360i)15-s + (−0.769 − 0.163i)16-s + (−0.801 − 0.356i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0649 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0649 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.0649 - 0.997i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.0649 - 0.997i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.24 - 1.57i)T \)
31 \( 1 + (4.78 - 2.85i)T \)
good2 \( 1 + (0.255 + 0.283i)T + (-0.209 + 1.98i)T^{2} \)
3 \( 1 + (2.18 + 0.464i)T + (2.74 + 1.22i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + (-0.313 + 2.98i)T + (-6.84 - 1.45i)T^{2} \)
11 \( 1 + (2.04 - 0.909i)T + (7.36 - 8.17i)T^{2} \)
17 \( 1 + (3.30 + 1.47i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (-5.86 + 1.24i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (-0.860 - 8.19i)T + (-22.4 + 4.78i)T^{2} \)
29 \( 1 + (2.26 + 2.51i)T + (-3.03 + 28.8i)T^{2} \)
37 \( 1 + (1.54 - 2.67i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.57 + 6.18i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (2.41 - 0.513i)T + (39.2 - 17.4i)T^{2} \)
47 \( 1 + (0.145 + 0.449i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.11 + 1.53i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-7.51 + 8.35i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (-4.28 - 7.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.88 - 6.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.3 + 5.47i)T + (47.5 + 52.7i)T^{2} \)
73 \( 1 + (8.85 + 6.43i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (8.42 - 6.12i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.83 - 5.65i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-4.43 + 1.97i)T + (59.5 - 66.1i)T^{2} \)
97 \( 1 + (-0.169 + 1.60i)T + (-94.8 - 20.1i)T^{2} \)
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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

−10.87143708131075409036402671140, −10.04733621616898491602733604790, −8.819976090378712038404637855244, −7.37634803892712998323904686471, −6.99234139832367516220855892529, −5.66558675364079297048971947518, −4.83409769174376064458904890213, −3.62585364725276172765604642927, −1.25837349847692517321067560713, 0, 2.94947077268940667191831812358, 4.10995565319241203093750224642, 5.26641125180554938714487943133, 6.21912708489748257667431903742, 7.33567426878200197213441043526, 8.325423764416979346667940242161, 8.816283464629413667778008619353, 10.46568064731824754111499698905, 11.36454869128791324371393347029

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