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Properties

Label 2-232-29.6-c1-0-1
Degree $2$
Conductor $232$
Sign $0.110 - 0.993i$
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  + (−1.15 − 0.264i)3-s + (−0.233 + 0.293i)5-s + (−0.933 + 4.08i)7-s + (−1.42 − 0.688i)9-s + (2.66 + 5.53i)11-s + (3.05 − 1.47i)13-s + (0.348 − 0.278i)15-s + 2.63i·17-s + (−2.82 + 0.644i)19-s + (2.16 − 4.49i)21-s + (−1.24 − 1.55i)23-s + (1.08 + 4.73i)25-s + (4.26 + 3.40i)27-s + (−1.04 + 5.28i)29-s + (−8.15 − 6.49i)31-s + ⋯
L(s)  = 1  + (−0.669 − 0.152i)3-s + (−0.104 + 0.131i)5-s + (−0.352 + 1.54i)7-s + (−0.476 − 0.229i)9-s + (0.804 + 1.67i)11-s + (0.847 − 0.408i)13-s + (0.0900 − 0.0718i)15-s + 0.639i·17-s + (−0.648 + 0.147i)19-s + (0.472 − 0.980i)21-s + (−0.258 − 0.324i)23-s + (0.216 + 0.947i)25-s + (0.820 + 0.654i)27-s + (−0.193 + 0.981i)29-s + (−1.46 − 1.16i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.110 - 0.993i$
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: \((2,\ 232,\ (\ :1/2),\ 0.110 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613212 + 0.548577i\)
\(L(\frac12)\) \(\approx\) \(0.613212 + 0.548577i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.04 - 5.28i)T \)
good3 \( 1 + (1.15 + 0.264i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (0.233 - 0.293i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.933 - 4.08i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-2.66 - 5.53i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-3.05 + 1.47i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 - 2.63iT - 17T^{2} \)
19 \( 1 + (2.82 - 0.644i)T + (17.1 - 8.24i)T^{2} \)
23 \( 1 + (1.24 + 1.55i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (8.15 + 6.49i)T + (6.89 + 30.2i)T^{2} \)
37 \( 1 + (-2.95 + 6.13i)T + (-23.0 - 28.9i)T^{2} \)
41 \( 1 + 4.75iT - 41T^{2} \)
43 \( 1 + (-8.25 + 6.58i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.679 - 1.41i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (1.50 - 1.88i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 2.09T + 59T^{2} \)
61 \( 1 + (-8.14 - 1.85i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (-3.15 - 1.51i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (8.41 - 4.05i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-1.92 + 1.53i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (-1.30 + 2.71i)T + (-49.2 - 61.7i)T^{2} \)
83 \( 1 + (-1.69 - 7.42i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-3.68 - 2.94i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + (-13.0 + 2.97i)T + (87.3 - 42.0i)T^{2} \)
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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

−12.56025051817479539704310532449, −11.56009659635733789556981513494, −10.69152175009567671896954360182, −9.316247377410188310401725517929, −8.755532131126368876343799913159, −7.22989720946231533617241572923, −6.12610107855487221219776936955, −5.46624924438994083278206437366, −3.82809450252462500963539326069, −2.10964487370200950338787193348, 0.75214892843366043442199099561, 3.40821971821102369401354209008, 4.44897155106177905525762568725, 5.96199399298490269495940272882, 6.66078607978165676452870189600, 8.044489942265451058993009452359, 9.017139227348702564853900223225, 10.31401839673167995746853173581, 11.14797178905393684386426194108, 11.56025239209141266002639706860

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