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Properties

Label 2-338-13.4-c1-0-0
Degree $2$
Conductor $338$
Sign $-0.839 - 0.543i$
Analytic cond. $2.69894$
Root an. cond. $1.64284$
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  + (−0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + 3i·5-s + (0.866 − 0.499i)6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (1 + 1.73i)9-s + (1.5 − 2.59i)10-s + (−5.19 − 3i)11-s − 0.999·12-s + 0.999·14-s + (−2.59 − 1.5i)15-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s − 2i·18-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + 1.34i·5-s + (0.353 − 0.204i)6-s + (−0.327 + 0.188i)7-s − 0.353i·8-s + (0.333 + 0.577i)9-s + (0.474 − 0.821i)10-s + (−1.56 − 0.904i)11-s − 0.288·12-s + 0.267·14-s + (−0.670 − 0.387i)15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s − 0.471i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ -0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.141684 + 0.479003i\)
\(L(\frac12)\) \(\approx\) \(0.141684 + 0.479003i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 + (0.866 - 0.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.19 + 3i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.73 - i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (-6.06 - 3.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (5.19 - 3i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.1 - 7i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.59 + 1.5i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (-5.19 - 3i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.66 + 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.40131911516793578677545286831, −10.75434082183712969071078777277, −10.41017309104638419011506509767, −9.395335635883032364636596366208, −8.123163309719125617923241976096, −7.31340679390919061922192814525, −6.20923662788991240261282913382, −4.98946712122626575731870419078, −3.34364686467396458963857996561, −2.46671400345628611632317300398, 0.41761814443316854574447251746, 2.03851173591619667733319079932, 4.26704193750054927992836800570, 5.35987344864234681401769977098, 6.39479735640052203940043672619, 7.51769705679523916301639526374, 8.216480056409014038341336623090, 9.335504685986561389336250012475, 9.974375550585275969288687687572, 11.10526878382598938105008692406

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