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Properties

Label 2-238-119.16-c1-0-5
Degree $2$
Conductor $238$
Sign $0.598 + 0.801i$
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  + (0.5 − 0.866i)2-s + (−2.59 + 1.5i)3-s + (−0.499 − 0.866i)4-s + 3i·6-s + (1.73 − 2i)7-s − 0.999·8-s + (3 − 5.19i)9-s + (2.59 − 1.5i)11-s + (2.59 + 1.50i)12-s + 5·13-s + (−0.866 − 2.5i)14-s + (−0.5 + 0.866i)16-s + (2.96 − 2.86i)17-s + (−3 − 5.19i)18-s + (−2 + 3.46i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−1.49 + 0.866i)3-s + (−0.249 − 0.433i)4-s + 1.22i·6-s + (0.654 − 0.755i)7-s − 0.353·8-s + (1 − 1.73i)9-s + (0.783 − 0.452i)11-s + (0.749 + 0.433i)12-s + 1.38·13-s + (−0.231 − 0.668i)14-s + (−0.125 + 0.216i)16-s + (0.718 − 0.695i)17-s + (−0.707 − 1.22i)18-s + (−0.458 + 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(238\)    =    \(2 \cdot 7 \cdot 17\)
Sign: $0.598 + 0.801i$
Analytic conductor: \(1.90043\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{238} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 238,\ (\ :1/2),\ 0.598 + 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.892969 - 0.447419i\)
\(L(\frac12)\) \(\approx\) \(0.892969 - 0.447419i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.73 + 2i)T \)
17 \( 1 + (-2.96 + 2.86i)T \)
good3 \( 1 + (2.59 - 1.5i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.46 + 2i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + (3.46 - 2i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - iT - 71T^{2} \)
73 \( 1 + (-6.92 + 4i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.06 + 3.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−11.61274471352311995540957570516, −11.21535787183644072235418045877, −10.38698604843270749199567803548, −9.622645489277903651620231647282, −8.168956086869568354640259688969, −6.40428611488213842166283134958, −5.73384178287771844887039889638, −4.42805155635050577051588752473, −3.80951710871072641624043892974, −1.04546580915508503420903568006, 1.59337003500745940107620069592, 4.13547027365039295748053769346, 5.53571914412022889265023203023, 5.98621119760161825965706189649, 7.03205899264091152436941442234, 8.020808056541671427869969667161, 9.196397664563957302152687705964, 10.88207100836535101467820317948, 11.46938311858907247435084636178, 12.34555669914710162187588431395

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