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Properties

Label 2-230-115.17-c1-0-3
Degree $2$
Conductor $230$
Sign $0.648 - 0.761i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
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  + (−0.936 + 0.349i)2-s + (1.26 − 0.691i)3-s + (0.755 − 0.654i)4-s + (−0.584 + 2.15i)5-s + (−0.944 + 1.09i)6-s + (1.35 + 1.80i)7-s + (−0.479 + 0.877i)8-s + (−0.496 + 0.773i)9-s + (−0.206 − 2.22i)10-s + (−0.475 + 0.217i)11-s + (0.504 − 1.35i)12-s + (3.11 + 2.32i)13-s + (−1.90 − 1.22i)14-s + (0.751 + 3.13i)15-s + (0.142 − 0.989i)16-s + (−0.578 − 8.08i)17-s + ⋯
L(s)  = 1  + (−0.662 + 0.247i)2-s + (0.730 − 0.399i)3-s + (0.377 − 0.327i)4-s + (−0.261 + 0.965i)5-s + (−0.385 + 0.445i)6-s + (0.511 + 0.683i)7-s + (−0.169 + 0.310i)8-s + (−0.165 + 0.257i)9-s + (−0.0652 − 0.704i)10-s + (−0.143 + 0.0654i)11-s + (0.145 − 0.390i)12-s + (0.863 + 0.646i)13-s + (−0.508 − 0.326i)14-s + (0.194 + 0.809i)15-s + (0.0355 − 0.247i)16-s + (−0.140 − 1.96i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.648 - 0.761i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.648 - 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02806 + 0.474848i\)
\(L(\frac12)\) \(\approx\) \(1.02806 + 0.474848i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.936 - 0.349i)T \)
5 \( 1 + (0.584 - 2.15i)T \)
23 \( 1 + (0.430 + 4.77i)T \)
good3 \( 1 + (-1.26 + 0.691i)T + (1.62 - 2.52i)T^{2} \)
7 \( 1 + (-1.35 - 1.80i)T + (-1.97 + 6.71i)T^{2} \)
11 \( 1 + (0.475 - 0.217i)T + (7.20 - 8.31i)T^{2} \)
13 \( 1 + (-3.11 - 2.32i)T + (3.66 + 12.4i)T^{2} \)
17 \( 1 + (0.578 + 8.08i)T + (-16.8 + 2.41i)T^{2} \)
19 \( 1 + (-4.90 - 5.66i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-1.93 - 1.67i)T + (4.12 + 28.7i)T^{2} \)
31 \( 1 + (0.910 - 0.267i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-2.51 + 0.546i)T + (33.6 - 15.3i)T^{2} \)
41 \( 1 + (5.19 - 3.34i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (4.84 + 8.86i)T + (-23.2 + 36.1i)T^{2} \)
47 \( 1 + (0.388 + 0.388i)T + 47iT^{2} \)
53 \( 1 + (-7.32 + 5.48i)T + (14.9 - 50.8i)T^{2} \)
59 \( 1 + (7.55 - 1.08i)T + (56.6 - 16.6i)T^{2} \)
61 \( 1 + (3.43 + 11.6i)T + (-51.3 + 32.9i)T^{2} \)
67 \( 1 + (3.30 + 8.86i)T + (-50.6 + 43.8i)T^{2} \)
71 \( 1 + (2.35 - 5.15i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-0.283 - 0.0202i)T + (72.2 + 10.3i)T^{2} \)
79 \( 1 + (-0.529 - 3.68i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (1.46 + 6.74i)T + (-75.4 + 34.4i)T^{2} \)
89 \( 1 + (-9.07 - 2.66i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (1.22 - 5.61i)T + (-88.2 - 40.2i)T^{2} \)
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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

−11.98323046525244300615702932474, −11.40345280972902021651537785746, −10.33724966753981248071795403162, −9.203548708419353482233873335165, −8.311355181512590429198490937537, −7.53721656427392872138169615206, −6.60130715649703933855735180148, −5.23070748482315376716246744596, −3.21111053515755037131443139486, −2.05725886595382197113136654700, 1.24253687831595956381206300499, 3.29673488039680282973564673581, 4.32729294557142124243976012675, 5.88971785767565487351266742195, 7.57603411841554202820194970319, 8.355801179115253818283666941495, 8.989168484775841761039141605994, 10.00934543391799523897547439709, 11.00625688710225646745296972940, 11.89083041284411398996024678999

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