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Properties

Label 2-234-117.32-c1-0-8
Degree $2$
Conductor $234$
Sign $0.656 - 0.754i$
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  + (0.707 + 0.707i)2-s + (1.72 − 0.0878i)3-s + 1.00i·4-s + (−2.64 + 0.707i)5-s + (1.28 + 1.16i)6-s + (3.36 − 0.902i)7-s + (−0.707 + 0.707i)8-s + (2.98 − 0.303i)9-s + (−2.36 − 1.36i)10-s + (−1.29 + 1.29i)11-s + (0.0878 + 1.72i)12-s + (2.77 + 2.29i)13-s + (3.01 + 1.74i)14-s + (−4.50 + 1.45i)15-s − 1.00·16-s + (−0.419 − 0.726i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.998 − 0.0507i)3-s + 0.500i·4-s + (−1.18 + 0.316i)5-s + (0.524 + 0.473i)6-s + (1.27 − 0.340i)7-s + (−0.250 + 0.250i)8-s + (0.994 − 0.101i)9-s + (−0.748 − 0.432i)10-s + (−0.389 + 0.389i)11-s + (0.0253 + 0.499i)12-s + (0.770 + 0.637i)13-s + (0.806 + 0.465i)14-s + (−1.16 + 0.375i)15-s − 0.250·16-s + (−0.101 − 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.656 - 0.754i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76937 + 0.806166i\)
\(L(\frac12)\) \(\approx\) \(1.76937 + 0.806166i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-1.72 + 0.0878i)T \)
13 \( 1 + (-2.77 - 2.29i)T \)
good5 \( 1 + (2.64 - 0.707i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-3.36 + 0.902i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.29 - 1.29i)T - 11iT^{2} \)
17 \( 1 + (0.419 + 0.726i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.94 + 1.86i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.18 + 5.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.05iT - 29T^{2} \)
31 \( 1 + (1.99 + 7.43i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.09 - 1.09i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.07 - 4.01i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-7.47 - 4.31i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.46 - 2.00i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 - 3.92iT - 53T^{2} \)
59 \( 1 + (5.26 - 5.26i)T - 59iT^{2} \)
61 \( 1 + (-0.247 + 0.428i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.34 - 1.43i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.48 + 5.55i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.66 + 9.66i)T + 73iT^{2} \)
79 \( 1 + (0.892 + 1.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.07 - 15.2i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.385 - 1.43i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.50 - 16.7i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.43440631593047181392952151273, −11.41124286123069041204551627465, −10.59804580090308258033108526103, −8.955976325421976565540661973571, −8.036682561711370426347950937832, −7.62668364499443227069868532478, −6.44019503729164085761689284523, −4.37569652384319929645317373701, −4.14401396106080470960231354727, −2.33450289998816887439109543694, 1.78011256973782064655196993641, 3.44153486310953352846277930990, 4.28645555823397683942387694710, 5.49623051706310724994070006952, 7.34358545763668899932339947467, 8.404918584834844251254146394898, 8.675579575201664504065788805421, 10.45456383442751010771666955922, 11.07932063357914598768337077306, 12.19280558585957473289216799568

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