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Properties

Label 2-234-117.110-c1-0-2
Degree $2$
Conductor $234$
Sign $-0.912 - 0.408i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
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  + (−0.258 + 0.965i)2-s + (−0.408 + 1.68i)3-s + (−0.866 − 0.499i)4-s + (−0.179 + 0.670i)5-s + (−1.52 − 0.830i)6-s + (3.24 + 3.24i)7-s + (0.707 − 0.707i)8-s + (−2.66 − 1.37i)9-s + (−0.601 − 0.347i)10-s + (−2.17 − 0.582i)11-s + (1.19 − 1.25i)12-s + (−1.78 + 3.13i)13-s + (−3.97 + 2.29i)14-s + (−1.05 − 0.576i)15-s + (0.500 + 0.866i)16-s + (−2.32 − 4.02i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.235 + 0.971i)3-s + (−0.433 − 0.249i)4-s + (−0.0803 + 0.299i)5-s + (−0.620 − 0.338i)6-s + (1.22 + 1.22i)7-s + (0.249 − 0.249i)8-s + (−0.888 − 0.458i)9-s + (−0.190 − 0.109i)10-s + (−0.655 − 0.175i)11-s + (0.345 − 0.361i)12-s + (−0.494 + 0.869i)13-s + (−1.06 + 0.613i)14-s + (−0.272 − 0.148i)15-s + (0.125 + 0.216i)16-s + (−0.563 − 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.912 - 0.408i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.912 - 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198643 + 0.929812i\)
\(L(\frac12)\) \(\approx\) \(0.198643 + 0.929812i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.408 - 1.68i)T \)
13 \( 1 + (1.78 - 3.13i)T \)
good5 \( 1 + (0.179 - 0.670i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-3.24 - 3.24i)T + 7iT^{2} \)
11 \( 1 + (2.17 + 0.582i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.32 + 4.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.81 + 0.753i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 4.74T + 23T^{2} \)
29 \( 1 + (4.22 - 2.44i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.51 - 1.74i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.44 + 0.924i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.05 - 4.05i)T + 41iT^{2} \)
43 \( 1 + 5.30iT - 43T^{2} \)
47 \( 1 + (-3.12 - 11.6i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 5.06iT - 53T^{2} \)
59 \( 1 + (1.38 + 5.18i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 + (-9.25 + 9.25i)T - 67iT^{2} \)
71 \( 1 + (-4.16 + 15.5i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.55 + 5.55i)T + 73iT^{2} \)
79 \( 1 + (1.55 - 2.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.17 + 2.45i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.648 - 2.41i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.868 - 0.868i)T - 97iT^{2} \)
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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

−12.42291125015898540934976498777, −11.31582249654976845742596231577, −10.83986507386436322254807593415, −9.327230946746666376295578351823, −8.888846092874052781680682297458, −7.74821033762167481132603681378, −6.40980741621482803762721287897, −5.15112221902451392247401102768, −4.66403129554790477549888486412, −2.64953148201147992451649359462, 0.894253536041374724430478855674, 2.36337806064464787129031730151, 4.26845057892379403766109563429, 5.34123237792018679787171315501, 6.97561865746962449109537665627, 7.961990424154430465584531986947, 8.473944534077374131853154591175, 10.25088535658457515310998101058, 10.86729108134887072590405959505, 11.66910477144612814869246223036

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