Properties

Label 2-232-232.109-c1-0-20
Degree $2$
Conductor $232$
Sign $-0.113 + 0.993i$
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  + (0.0855 + 1.41i)2-s + (−1.28 − 0.619i)3-s + (−1.98 + 0.241i)4-s + (−1.62 + 0.370i)5-s + (0.763 − 1.86i)6-s + (0.720 + 0.346i)7-s + (−0.510 − 2.78i)8-s + (−0.600 − 0.753i)9-s + (−0.662 − 2.26i)10-s + (0.468 − 0.586i)11-s + (2.70 + 0.918i)12-s + (−5.21 − 4.15i)13-s + (−0.428 + 1.04i)14-s + (2.31 + 0.529i)15-s + (3.88 − 0.959i)16-s − 6.16i·17-s + ⋯
L(s)  = 1  + (0.0604 + 0.998i)2-s + (−0.742 − 0.357i)3-s + (−0.992 + 0.120i)4-s + (−0.726 + 0.165i)5-s + (0.311 − 0.762i)6-s + (0.272 + 0.131i)7-s + (−0.180 − 0.983i)8-s + (−0.200 − 0.251i)9-s + (−0.209 − 0.715i)10-s + (0.141 − 0.176i)11-s + (0.780 + 0.265i)12-s + (−1.44 − 1.15i)13-s + (−0.114 + 0.279i)14-s + (0.598 + 0.136i)15-s + (0.970 − 0.239i)16-s − 1.49i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $-0.113 + 0.993i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ -0.113 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.157527 - 0.176543i\)
\(L(\frac12)\) \(\approx\) \(0.157527 - 0.176543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0855 - 1.41i)T \)
29 \( 1 + (-2.43 - 4.80i)T \)
good3 \( 1 + (1.28 + 0.619i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (1.62 - 0.370i)T + (4.50 - 2.16i)T^{2} \)
7 \( 1 + (-0.720 - 0.346i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-0.468 + 0.586i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (5.21 + 4.15i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + 6.16iT - 17T^{2} \)
19 \( 1 + (1.97 - 0.952i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (0.962 - 4.21i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (7.61 - 1.73i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (0.414 + 0.519i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 0.363iT - 41T^{2} \)
43 \( 1 + (-1.89 + 8.29i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-3.94 - 3.14i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.146 - 0.0335i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + 4.94iT - 59T^{2} \)
61 \( 1 + (1.37 + 0.663i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (-6.17 + 4.92i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (0.582 - 0.730i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (12.1 + 2.78i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (8.96 - 7.14i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-2.44 - 5.08i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (-6.33 + 1.44i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (2.10 + 4.37i)T + (-60.4 + 75.8i)T^{2} \)
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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

−12.12359883987140491021254332845, −11.17225920467902396732804366286, −9.858635015156387230217892969296, −8.778854949921937905292080918142, −7.53863769552560740836609429815, −7.09351787322890916460044179864, −5.67920127464087590801754708278, −4.99246068705598446344919865581, −3.37845868534857449052619411514, −0.20192058667269735872170014972, 2.19858078986285683060559562103, 4.16688684741234370214788116986, 4.65512361460954449514992501118, 6.05542233910610852194581016661, 7.69820834259756676452033423424, 8.706795640187270477372806731002, 9.891281851765024566808708403145, 10.70203323315877195726198254124, 11.55551355804468593269492001478, 12.11042657761771834027294229535

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