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Properties

Label 2-633-633.632-c1-0-9
Degree $2$
Conductor $633$
Sign $0.605 - 0.796i$
Analytic cond. $5.05453$
Root an. cond. $2.24822$
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  − 1.55·2-s + (−1.58 − 0.697i)3-s + 0.404·4-s + 3.05i·5-s + (2.45 + 1.08i)6-s − 1.87i·7-s + 2.47·8-s + (2.02 + 2.21i)9-s − 4.73i·10-s − 3.63i·11-s + (−0.640 − 0.281i)12-s − 3.56·13-s + 2.90i·14-s + (2.13 − 4.84i)15-s − 4.64·16-s − 2.71·17-s + ⋯
L(s)  = 1  − 1.09·2-s + (−0.915 − 0.402i)3-s + 0.202·4-s + 1.36i·5-s + (1.00 + 0.441i)6-s − 0.706i·7-s + 0.874·8-s + (0.675 + 0.737i)9-s − 1.49i·10-s − 1.09i·11-s + (−0.184 − 0.0813i)12-s − 0.988·13-s + 0.775i·14-s + (0.549 − 1.25i)15-s − 1.16·16-s − 0.657·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $0.605 - 0.796i$
Analytic conductor: \(5.05453\)
Root analytic conductor: \(2.24822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{633} (632, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 633,\ (\ :1/2),\ 0.605 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.395262 + 0.196039i\)
\(L(\frac12)\) \(\approx\) \(0.395262 + 0.196039i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.58 + 0.697i)T \)
211 \( 1 + (-3.39 + 14.1i)T \)
good2 \( 1 + 1.55T + 2T^{2} \)
5 \( 1 - 3.05iT - 5T^{2} \)
7 \( 1 + 1.87iT - 7T^{2} \)
11 \( 1 + 3.63iT - 11T^{2} \)
13 \( 1 + 3.56T + 13T^{2} \)
17 \( 1 + 2.71T + 17T^{2} \)
19 \( 1 - 5.44T + 19T^{2} \)
23 \( 1 + 1.21T + 23T^{2} \)
29 \( 1 + 1.12T + 29T^{2} \)
31 \( 1 - 7.12iT - 31T^{2} \)
37 \( 1 - 4.14T + 37T^{2} \)
41 \( 1 - 3.56T + 41T^{2} \)
43 \( 1 - 8.68T + 43T^{2} \)
47 \( 1 - 2.43iT - 47T^{2} \)
53 \( 1 - 5.03iT - 53T^{2} \)
59 \( 1 + 2.00iT - 59T^{2} \)
61 \( 1 - 7.72iT - 61T^{2} \)
67 \( 1 - 10.4iT - 67T^{2} \)
71 \( 1 - 6.67iT - 71T^{2} \)
73 \( 1 + 8.63T + 73T^{2} \)
79 \( 1 - 1.49T + 79T^{2} \)
83 \( 1 - 3.53iT - 83T^{2} \)
89 \( 1 - 17.9T + 89T^{2} \)
97 \( 1 - 18.3iT - 97T^{2} \)
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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

−10.67932237995690876848633038660, −10.08789790352947551780748903637, −9.120363331301891916159909328195, −7.78662412528247497428953764196, −7.31183836038313981645888136493, −6.59998538830725598367815560582, −5.46032932789007185704473161799, −4.17602636895282130010332394692, −2.65557140531363720953874316848, −0.961068820773443163634876895100, 0.53469025671491312128037385930, 1.96616695930868741210013970475, 4.34577654015238384808776782236, 4.86984947223669553799952203063, 5.76587051268155465379073623746, 7.18722124612832983788563353958, 7.936738481478493640731234007946, 9.250298449084192448073254589117, 9.336139979830440907887611885234, 10.11845866051903940503393265975

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