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Properties

Label 2-234-117.2-c1-0-3
Degree $2$
Conductor $234$
Sign $-0.00876 - 0.999i$
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 + (−0.613 − 1.61i)3-s + 1.00i·4-s + (−0.964 + 3.60i)5-s + (0.711 − 1.57i)6-s + (−0.861 + 3.21i)7-s + (−0.707 + 0.707i)8-s + (−2.24 + 1.98i)9-s + (−3.22 + 1.86i)10-s + (4.48 − 4.48i)11-s + (1.61 − 0.613i)12-s + (−0.905 + 3.49i)13-s + (−2.88 + 1.66i)14-s + (6.42 − 0.648i)15-s − 1.00·16-s + (1.38 − 2.39i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.354 − 0.935i)3-s + 0.500i·4-s + (−0.431 + 1.61i)5-s + (0.290 − 0.644i)6-s + (−0.325 + 1.21i)7-s + (−0.250 + 0.250i)8-s + (−0.748 + 0.662i)9-s + (−1.02 + 0.589i)10-s + (1.35 − 1.35i)11-s + (0.467 − 0.177i)12-s + (−0.251 + 0.967i)13-s + (−0.769 + 0.444i)14-s + (1.65 − 0.167i)15-s − 0.250·16-s + (0.335 − 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00876 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00876 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.00876 - 0.999i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.00876 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.872204 + 0.879882i\)
\(L(\frac12)\) \(\approx\) \(0.872204 + 0.879882i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.613 + 1.61i)T \)
13 \( 1 + (0.905 - 3.49i)T \)
good5 \( 1 + (0.964 - 3.60i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.861 - 3.21i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.48 + 4.48i)T - 11iT^{2} \)
17 \( 1 + (-1.38 + 2.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.703 - 2.62i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.0915 + 0.158i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.948iT - 29T^{2} \)
31 \( 1 + (-2.14 - 0.573i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.96 + 7.33i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.27 + 0.610i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.17 - 1.83i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.38 + 5.15i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 3.43iT - 53T^{2} \)
59 \( 1 + (-3.31 + 3.31i)T - 59iT^{2} \)
61 \( 1 + (-4.13 - 7.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.93 - 7.21i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-11.7 + 3.16i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.16 - 2.16i)T + 73iT^{2} \)
79 \( 1 + (-6.31 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.52 + 1.74i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.643 - 0.172i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.65 + 1.51i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.08837343931420547857368764358, −11.77075306026482519949505709547, −10.99038061654535944481839249357, −9.297294113412895050076206202471, −8.200628850035238480424338504284, −7.03704804068515012501950382194, −6.40813519330399478083566002348, −5.68958889035901315550257155739, −3.63454551752554524597578947707, −2.51966181157605919333886044410, 0.981952419270177979770686797286, 3.69176185283409599137163280790, 4.42015722236164268650897843812, 5.16298779266482075905630000785, 6.67728003375373750225445041916, 8.166029560064842155365417978459, 9.467605221972951453006356866619, 9.900184557573710356318236125110, 11.05227079311163985671418501624, 12.10949652696952924596644178340

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