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Properties

Label 2-234-117.11-c1-0-13
Degree $2$
Conductor $234$
Sign $-0.986 + 0.165i$
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 + (−1.25 − 1.19i)3-s − 1.00i·4-s + (−0.670 − 0.179i)5-s + (−1.73 + 0.0409i)6-s + (−4.43 − 1.18i)7-s + (−0.707 − 0.707i)8-s + (0.141 + 2.99i)9-s + (−0.601 + 0.347i)10-s + (−1.59 − 1.59i)11-s + (−1.19 + 1.25i)12-s + (3.60 + 0.0239i)13-s + (−3.97 + 2.29i)14-s + (0.625 + 1.02i)15-s − 1.00·16-s + (2.32 − 4.02i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.723 − 0.690i)3-s − 0.500i·4-s + (−0.299 − 0.0803i)5-s + (−0.706 + 0.0167i)6-s + (−1.67 − 0.449i)7-s + (−0.250 − 0.250i)8-s + (0.0473 + 0.998i)9-s + (−0.190 + 0.109i)10-s + (−0.479 − 0.479i)11-s + (−0.345 + 0.361i)12-s + (0.999 + 0.00665i)13-s + (−1.06 + 0.613i)14-s + (0.161 + 0.265i)15-s − 0.250·16-s + (0.563 − 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.986 + 0.165i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.986 + 0.165i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0629357 - 0.753915i\)
\(L(\frac12)\) \(\approx\) \(0.0629357 - 0.753915i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 + 1.19i)T \)
13 \( 1 + (-3.60 - 0.0239i)T \)
good5 \( 1 + (0.670 + 0.179i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (4.43 + 1.18i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.59 + 1.59i)T + 11iT^{2} \)
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 \( 1 + (-2.37 + 4.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.88iT - 29T^{2} \)
31 \( 1 + (1.74 - 6.51i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.44 - 0.924i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.48 + 5.54i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-4.59 + 2.65i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-11.6 + 3.12i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 5.06iT - 53T^{2} \)
59 \( 1 + (-3.79 - 3.79i)T + 59iT^{2} \)
61 \( 1 + (1.56 + 2.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.6 - 3.38i)T + (58.0 - 33.5i)T^{2} \)
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 \( 1 + (-2.45 - 9.17i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.648 - 2.41i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.317 - 1.18i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.96882066816761200640942075546, −10.79534639494219081444004614593, −10.23316030943464985236522109899, −8.868881636025087188413331922286, −7.43210069128308960580559921666, −6.41657287840296855911874684157, −5.66632252230353468995689670788, −4.09263967220680990316266650833, −2.79855920146979738037193000879, −0.56410138344717579706925277596, 3.24420077702307549269926172752, 4.13070787457407539071603550402, 5.70711127276821269586425292053, 6.18297186000418104276401650558, 7.38124794852982555505405861556, 8.870657936626562857157593481008, 9.751376553025913507387061790114, 10.74192573937188802771206888837, 11.77648176326977736820218311168, 12.81809831417583531794254530688

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