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Properties

Label 2-238-7.4-c1-0-5
Degree $2$
Conductor $238$
Sign $0.386 - 0.922i$
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 + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (1 + 1.73i)5-s + 0.999·6-s + (−2 + 1.73i)7-s − 0.999·8-s + (1 + 1.73i)9-s + (−0.999 + 1.73i)10-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s + 5·13-s + (−2.5 − 0.866i)14-s + 1.99·15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.447 + 0.774i)5-s + 0.408·6-s + (−0.755 + 0.654i)7-s − 0.353·8-s + (0.333 + 0.577i)9-s + (−0.316 + 0.547i)10-s + (0.150 − 0.261i)11-s + (0.144 + 0.249i)12-s + 1.38·13-s + (−0.668 − 0.231i)14-s + 0.516·15-s + (−0.125 − 0.216i)16-s + (0.121 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(238\)    =    \(2 \cdot 7 \cdot 17\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(1.90043\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{238} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 238,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34081 + 0.891884i\)
\(L(\frac12)\) \(\approx\) \(1.34081 + 0.891884i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.5 + 6.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−12.79348523722218745012008879074, −11.38568417233479250575594470800, −10.45638914810502961483803892897, −9.155065888105107479082298857683, −8.354900601076673600442950202119, −7.01319826621547536290420362109, −6.44455182547807385835845326024, −5.35512925343280538803065050888, −3.61751441304990806598001717871, −2.34862110339716896619279711302, 1.41122341362382120862116286465, 3.52567205019278328548120295689, 4.15259246698193771552591432243, 5.65351290544615791559171850687, 6.68000034832818214163693126838, 8.389557331501682570663053311767, 9.335594379323794776518501887713, 10.00201167277202756392568399579, 10.86632616735349538675347233578, 12.14889593285371374490426980238

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