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Properties

Label 2-232-29.6-c1-0-7
Degree $2$
Conductor $232$
Sign $-0.773 + 0.633i$
Analytic cond. $1.85252$
Root an. cond. $1.36107$
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  + (−1.64 − 0.375i)3-s + (−0.945 + 1.18i)5-s + (0.445 − 1.95i)7-s + (−0.132 − 0.0637i)9-s + (−2.56 − 5.32i)11-s + (−3.47 + 1.67i)13-s + (2.00 − 1.59i)15-s − 2.92i·17-s + (−4.67 + 1.06i)19-s + (−1.46 + 3.04i)21-s + (−2.78 − 3.49i)23-s + (0.600 + 2.63i)25-s + (4.15 + 3.31i)27-s + (4.43 − 3.05i)29-s + (1.99 + 1.59i)31-s + ⋯
L(s)  = 1  + (−0.950 − 0.217i)3-s + (−0.423 + 0.530i)5-s + (0.168 − 0.737i)7-s + (−0.0441 − 0.0212i)9-s + (−0.772 − 1.60i)11-s + (−0.963 + 0.463i)13-s + (0.517 − 0.412i)15-s − 0.709i·17-s + (−1.07 + 0.244i)19-s + (−0.320 + 0.664i)21-s + (−0.581 − 0.728i)23-s + (0.120 + 0.526i)25-s + (0.799 + 0.637i)27-s + (0.823 − 0.566i)29-s + (0.359 + 0.286i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $-0.773 + 0.633i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ -0.773 + 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.125136 - 0.350464i\)
\(L(\frac12)\) \(\approx\) \(0.125136 - 0.350464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-4.43 + 3.05i)T \)
good3 \( 1 + (1.64 + 0.375i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (0.945 - 1.18i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (-0.445 + 1.95i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (2.56 + 5.32i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (3.47 - 1.67i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 2.92iT - 17T^{2} \)
19 \( 1 + (4.67 - 1.06i)T + (17.1 - 8.24i)T^{2} \)
23 \( 1 + (2.78 + 3.49i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-1.99 - 1.59i)T + (6.89 + 30.2i)T^{2} \)
37 \( 1 + (-0.355 + 0.737i)T + (-23.0 - 28.9i)T^{2} \)
41 \( 1 + 8.28iT - 41T^{2} \)
43 \( 1 + (1.04 - 0.830i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-2.15 - 4.46i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (1.94 - 2.44i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 7.30T + 59T^{2} \)
61 \( 1 + (-3.52 - 0.805i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (5.55 + 2.67i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (1.63 - 0.788i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (8.52 - 6.79i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (-6.17 + 12.8i)T + (-49.2 - 61.7i)T^{2} \)
83 \( 1 + (-3.25 - 14.2i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (9.99 + 7.97i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + (-8.03 + 1.83i)T + (87.3 - 42.0i)T^{2} \)
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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

−11.67493168623153914900304420037, −10.90921591492676205106940972479, −10.34336299802810659422937347414, −8.757186837751782854717995414188, −7.66260064896716361034008199588, −6.67814336543083397223932700694, −5.71623912837119737656737249287, −4.44744848739027747723118561548, −2.90698046601660273433026518906, −0.32231055254011606618228058711, 2.35535434641258422706385710727, 4.54450959968060560617472778785, 5.12289460151530795773469544730, 6.30638444303617807200367550982, 7.67406749725790406398804963149, 8.566024595545531493284931942520, 9.932154124136718269547790904830, 10.59215864073392580230912373339, 11.89317169099747481786442601150, 12.27855448947714581490473125346

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