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Properties

Label 2-238-119.48-c1-0-2
Degree $2$
Conductor $238$
Sign $0.492 - 0.870i$
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 + (0.596 + 0.118i)3-s + (0.707 − 0.707i)4-s + (1.74 + 2.61i)5-s + (−0.596 + 0.118i)6-s + (2.56 + 0.668i)7-s + (−0.382 + 0.923i)8-s + (−2.43 − 1.00i)9-s + (−2.61 − 1.74i)10-s + (−3.66 + 0.728i)11-s + (0.505 − 0.337i)12-s + (2.81 − 2.81i)13-s + (−2.62 + 0.362i)14-s + (0.732 + 1.76i)15-s − i·16-s + (2.54 + 3.24i)17-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (0.344 + 0.0684i)3-s + (0.353 − 0.353i)4-s + (0.782 + 1.17i)5-s + (−0.243 + 0.0484i)6-s + (0.967 + 0.252i)7-s + (−0.135 + 0.326i)8-s + (−0.810 − 0.335i)9-s + (−0.828 − 0.553i)10-s + (−1.10 + 0.219i)11-s + (0.145 − 0.0974i)12-s + (0.779 − 0.779i)13-s + (−0.700 + 0.0968i)14-s + (0.189 + 0.456i)15-s − 0.250i·16-s + (0.617 + 0.786i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(238\)    =    \(2 \cdot 7 \cdot 17\)
Sign: $0.492 - 0.870i$
Analytic conductor: \(1.90043\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{238} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 238,\ (\ :1/2),\ 0.492 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00847 + 0.588376i\)
\(L(\frac12)\) \(\approx\) \(1.00847 + 0.588376i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 - 0.668i)T \)
17 \( 1 + (-2.54 - 3.24i)T \)
good3 \( 1 + (-0.596 - 0.118i)T + (2.77 + 1.14i)T^{2} \)
5 \( 1 + (-1.74 - 2.61i)T + (-1.91 + 4.61i)T^{2} \)
11 \( 1 + (3.66 - 0.728i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (-2.81 + 2.81i)T - 13iT^{2} \)
19 \( 1 + (-2.08 - 5.04i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.0421 - 0.211i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (1.71 + 2.57i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (-0.521 + 2.62i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-0.603 + 3.03i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-4.30 + 6.43i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (3.97 + 1.64i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (3.26 - 3.26i)T - 47iT^{2} \)
53 \( 1 + (-3.59 + 1.48i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (7.33 + 3.03i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.63 + 1.09i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + 15.5iT - 67T^{2} \)
71 \( 1 + (-1.58 + 7.96i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (6.68 + 10.0i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (-0.195 + 0.0389i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (13.2 - 5.48i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-7.68 - 7.68i)T + 89iT^{2} \)
97 \( 1 + (-0.729 + 0.487i)T + (37.1 - 89.6i)T^{2} \)
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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

−12.12185033984933600633526052416, −10.88819564089033172657753451475, −10.48417940727001804311405240145, −9.436712613502918517927869156091, −8.166569176961008577426731942001, −7.71540366672928893536092476120, −6.09179087074153221275139003095, −5.55408123215993211038219084779, −3.29202756644889665003412669658, −2.05606816020882484118004020130, 1.32300259502142410338589524936, 2.75779266756742961736773980627, 4.77769988999942945226376217285, 5.62143125327690595744660979653, 7.34310939821688868954395284424, 8.414058713222931204729842455911, 8.866089021938687052237871664022, 9.906773748208894916566445849231, 11.09447561991675961181052369093, 11.70396737374768886082135006098

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