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Properties

Label 2-633-633.632-c1-0-66
Degree $2$
Conductor $633$
Sign $0.177 + 0.984i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $0.177 + 0.984i$
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: \((2,\ 633,\ (\ :1/2),\ 0.177 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30915 - 1.92940i\)
\(L(\frac12)\) \(\approx\) \(2.30915 - 1.92940i\)
\(L(\frac{3}{2})\) 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

−10.09321567910573105352391570849, −9.455013265226791678292799798268, −8.705615016198987825158983211294, −7.62377350972645333685792434087, −6.96532217727486531560043939039, −5.44423455461980116875765384962, −4.70423542910849909367054159295, −3.95653548621479580769536126489, −2.75338383553079598404119277262, −1.17633993791956698581685490515, 2.67100099518338928608038569651, 3.00433605813974815659101708460, 4.01308440657348692601086591424, 5.25499233625955869103232349056, 6.02535634200288351926832084090, 7.20554322087472051686354921891, 8.067369365121994078315478747652, 9.226949426076477811802619588197, 9.794231791830804401676814999491, 10.89932793570899476652344656920

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