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Properties

Label 2-238-119.6-c1-0-6
Degree $2$
Conductor $238$
Sign $0.0868 - 0.996i$
Analytic cond. $1.90043$
Root an. cond. $1.37856$
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  + (0.923 + 0.382i)2-s + (0.459 + 2.31i)3-s + (0.707 + 0.707i)4-s + (1.50 + 1.00i)5-s + (−0.459 + 2.31i)6-s + (−1.43 − 2.22i)7-s + (0.382 + 0.923i)8-s + (−2.35 + 0.975i)9-s + (1.00 + 1.50i)10-s + (0.483 − 2.42i)11-s + (−1.30 + 1.95i)12-s + (−0.638 − 0.638i)13-s + (−0.476 − 2.60i)14-s + (−1.63 + 3.94i)15-s + i·16-s + (−3.66 − 1.88i)17-s + ⋯
L(s)  = 1  + (0.653 + 0.270i)2-s + (0.265 + 1.33i)3-s + (0.353 + 0.353i)4-s + (0.674 + 0.450i)5-s + (−0.187 + 0.943i)6-s + (−0.542 − 0.839i)7-s + (0.135 + 0.326i)8-s + (−0.785 + 0.325i)9-s + (0.318 + 0.476i)10-s + (0.145 − 0.732i)11-s + (−0.377 + 0.565i)12-s + (−0.177 − 0.177i)13-s + (−0.127 − 0.695i)14-s + (−0.421 + 1.01i)15-s + 0.250i·16-s + (−0.889 − 0.456i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0868 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0868 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(238\)    =    \(2 \cdot 7 \cdot 17\)
Sign: $0.0868 - 0.996i$
Analytic conductor: \(1.90043\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{238} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 238,\ (\ :1/2),\ 0.0868 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44469 + 1.32422i\)
\(L(\frac12)\) \(\approx\) \(1.44469 + 1.32422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 + (1.43 + 2.22i)T \)
17 \( 1 + (3.66 + 1.88i)T \)
good3 \( 1 + (-0.459 - 2.31i)T + (-2.77 + 1.14i)T^{2} \)
5 \( 1 + (-1.50 - 1.00i)T + (1.91 + 4.61i)T^{2} \)
11 \( 1 + (-0.483 + 2.42i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (0.638 + 0.638i)T + 13iT^{2} \)
19 \( 1 + (0.0984 - 0.237i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.154 + 0.0308i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (-8.40 - 5.61i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (5.96 - 1.18i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (-2.80 + 0.558i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (-8.45 + 5.64i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (0.396 - 0.164i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + (7.12 + 7.12i)T + 47iT^{2} \)
53 \( 1 + (1.54 + 0.638i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (7.72 - 3.20i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-2.67 - 4.00i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 + (11.0 - 2.19i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-7.21 - 4.81i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (2.07 - 10.4i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (2.10 + 0.873i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-8.43 + 8.43i)T - 89iT^{2} \)
97 \( 1 + (6.63 - 9.92i)T + (-37.1 - 89.6i)T^{2} \)
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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

−12.55712352680385976196723607873, −11.07874706180679306521110983717, −10.50089362072410776664249614080, −9.636820652514433743041164551117, −8.658759416559283115415091840023, −7.12151149152176112324534673144, −6.16220511296294111366943472372, −4.91355713430687844894444473666, −3.87096822611489124058725650239, −2.86935918468407535104659521830, 1.71640040598258246447170550704, 2.65591171040544258715903555784, 4.58119046844845174451840111967, 5.98303456085072172679617321389, 6.59899849487424835337756848911, 7.81399249445871717852782718821, 9.043013282903278521407055607718, 9.883988220704130515010923828814, 11.37773937275589579179190273934, 12.33725427159257117332659634232

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