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Properties

Label 2-232-232.109-c1-0-13
Degree $2$
Conductor $232$
Sign $0.975 - 0.218i$
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  + (−0.167 + 1.40i)2-s + (−1.07 − 0.516i)3-s + (−1.94 − 0.469i)4-s + (3.71 − 0.848i)5-s + (0.904 − 1.42i)6-s + (−1.24 − 0.599i)7-s + (0.983 − 2.65i)8-s + (−0.986 − 1.23i)9-s + (0.570 + 5.36i)10-s + (3.28 − 4.12i)11-s + (1.84 + 1.50i)12-s + (3.13 + 2.49i)13-s + (1.04 − 1.64i)14-s + (−4.42 − 1.01i)15-s + (3.55 + 1.82i)16-s + 5.15i·17-s + ⋯
L(s)  = 1  + (−0.118 + 0.993i)2-s + (−0.619 − 0.298i)3-s + (−0.972 − 0.234i)4-s + (1.66 − 0.379i)5-s + (0.369 − 0.579i)6-s + (−0.470 − 0.226i)7-s + (0.347 − 0.937i)8-s + (−0.328 − 0.412i)9-s + (0.180 + 1.69i)10-s + (0.991 − 1.24i)11-s + (0.532 + 0.435i)12-s + (0.868 + 0.692i)13-s + (0.280 − 0.440i)14-s + (−1.14 − 0.260i)15-s + (0.889 + 0.456i)16-s + 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.975 - 0.218i$
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: \((2,\ 232,\ (\ :1/2),\ 0.975 - 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09377 + 0.121194i\)
\(L(\frac12)\) \(\approx\) \(1.09377 + 0.121194i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.167 - 1.40i)T \)
29 \( 1 + (-1.22 + 5.24i)T \)
good3 \( 1 + (1.07 + 0.516i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-3.71 + 0.848i)T + (4.50 - 2.16i)T^{2} \)
7 \( 1 + (1.24 + 0.599i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-3.28 + 4.12i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-3.13 - 2.49i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 - 5.15iT - 17T^{2} \)
19 \( 1 + (0.763 - 0.367i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.615 + 2.69i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (1.58 - 0.362i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (5.69 + 7.14i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 - 3.88iT - 41T^{2} \)
43 \( 1 + (1.59 - 6.97i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (1.40 + 1.12i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (-0.753 + 0.171i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 + (-0.857 - 0.412i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (-9.70 + 7.74i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (3.97 - 4.98i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (1.03 + 0.235i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (11.3 - 9.05i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-3.13 - 6.51i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (6.63 - 1.51i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-6.84 - 14.2i)T + (-60.4 + 75.8i)T^{2} \)
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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

−12.56762892780138424597009572686, −11.16366847956605637820177584970, −10.06121014012800173611908441989, −9.059935043803982390879176743420, −8.550638161384750295247618183601, −6.47165001607606430234062796671, −6.33337275851143518209791629360, −5.53703966554486423936438406432, −3.84665752518656833444907619010, −1.22129761073357472180752070135, 1.77458505204574209354771832328, 3.10464507390897999351471786710, 4.88984354390710348498499411076, 5.71878227952055926454814309581, 6.91151783305254355153375503288, 8.770529221490128657770565809756, 9.630834167799549002516232370291, 10.20217670554379259763874360486, 11.06096281785754983745894192071, 12.03633847868366422592568616811

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