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Properties

Label 2-272-272.13-c1-0-15
Degree $2$
Conductor $272$
Sign $-0.355 - 0.934i$
Analytic cond. $2.17193$
Root an. cond. $1.47374$
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  + (0.930 + 1.06i)2-s + 1.70i·3-s + (−0.268 + 1.98i)4-s + 2.90·5-s + (−1.81 + 1.58i)6-s + (−0.453 − 0.453i)7-s + (−2.36 + 1.55i)8-s + 0.109·9-s + (2.70 + 3.09i)10-s − 2.30i·11-s + (−3.36 − 0.455i)12-s + (−2.83 − 2.83i)13-s + (0.0608 − 0.904i)14-s + 4.93i·15-s + (−3.85 − 1.06i)16-s + (2.55 − 3.23i)17-s + ⋯
L(s)  = 1  + (0.658 + 0.753i)2-s + 0.981i·3-s + (−0.134 + 0.990i)4-s + 1.29·5-s + (−0.739 + 0.645i)6-s + (−0.171 − 0.171i)7-s + (−0.834 + 0.551i)8-s + 0.0363·9-s + (0.854 + 0.977i)10-s − 0.693i·11-s + (−0.972 − 0.131i)12-s + (−0.784 − 0.784i)13-s + (0.0162 − 0.241i)14-s + 1.27i·15-s + (−0.964 − 0.265i)16-s + (0.619 − 0.784i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.355 - 0.934i$
Analytic conductor: \(2.17193\)
Root analytic conductor: \(1.47374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :1/2),\ -0.355 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12992 + 1.63815i\)
\(L(\frac12)\) \(\approx\) \(1.12992 + 1.63815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.930 - 1.06i)T \)
17 \( 1 + (-2.55 + 3.23i)T \)
good3 \( 1 - 1.70iT - 3T^{2} \)
5 \( 1 - 2.90T + 5T^{2} \)
7 \( 1 + (0.453 + 0.453i)T + 7iT^{2} \)
11 \( 1 + 2.30iT - 11T^{2} \)
13 \( 1 + (2.83 + 2.83i)T + 13iT^{2} \)
19 \( 1 + (2.28 - 2.28i)T - 19iT^{2} \)
23 \( 1 + (2.63 + 2.63i)T + 23iT^{2} \)
29 \( 1 - 3.29iT - 29T^{2} \)
31 \( 1 + (3.92 + 3.92i)T + 31iT^{2} \)
37 \( 1 - 1.27T + 37T^{2} \)
41 \( 1 + (-4.47 - 4.47i)T + 41iT^{2} \)
43 \( 1 + (-4.30 - 4.30i)T + 43iT^{2} \)
47 \( 1 - 5.63T + 47T^{2} \)
53 \( 1 + (4.23 + 4.23i)T + 53iT^{2} \)
59 \( 1 + (3.50 + 3.50i)T + 59iT^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + (-8.74 - 8.74i)T + 67iT^{2} \)
71 \( 1 + (-0.687 + 0.687i)T - 71iT^{2} \)
73 \( 1 + (11.8 - 11.8i)T - 73iT^{2} \)
79 \( 1 + (7.30 - 7.30i)T - 79iT^{2} \)
83 \( 1 + (4.42 - 4.42i)T - 83iT^{2} \)
89 \( 1 - 0.595iT - 89T^{2} \)
97 \( 1 + (9.56 + 9.56i)T + 97iT^{2} \)
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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

−12.59168611654171963619124840123, −11.17777110899382489062491985610, −10.03439621496113395994694438035, −9.539170119512114320767971572119, −8.329035731743863103267248853625, −7.10378492024489053879626412036, −5.86342174408225578799421231055, −5.25960360157069298455405059227, −4.04020956851111282343698455671, −2.73546140916818216595477936555, 1.66297313200089996657726647683, 2.38757093497649230230057524505, 4.27613447959102256989990576752, 5.60773210642134508110549948266, 6.41372093827947164237668093390, 7.40485495381905089456053534443, 9.143283510665501005788987819465, 9.827026045536740911017991603991, 10.68176386017358614509720845989, 12.07645871313715443490487831671

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