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Properties

Label 2-1380-92.91-c1-0-30
Degree $2$
Conductor $1380$
Sign $0.347 - 0.937i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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  + (−0.651 + 1.25i)2-s − i·3-s + (−1.15 − 1.63i)4-s + i·5-s + (1.25 + 0.651i)6-s − 2.14·7-s + (2.80 − 0.381i)8-s − 9-s + (−1.25 − 0.651i)10-s + 3.55·11-s + (−1.63 + 1.15i)12-s − 4.04·13-s + (1.39 − 2.68i)14-s + 15-s + (−1.34 + 3.76i)16-s + 0.843i·17-s + ⋯
L(s)  = 1  + (−0.460 + 0.887i)2-s − 0.577i·3-s + (−0.575 − 0.817i)4-s + 0.447i·5-s + (0.512 + 0.265i)6-s − 0.808·7-s + (0.990 − 0.134i)8-s − 0.333·9-s + (−0.396 − 0.205i)10-s + 1.07·11-s + (−0.471 + 0.332i)12-s − 1.12·13-s + (0.372 − 0.718i)14-s + 0.258·15-s + (−0.336 + 0.941i)16-s + 0.204i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.347 - 0.937i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.347 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.007438645\)
\(L(\frac12)\) \(\approx\) \(1.007438645\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.651 - 1.25i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
23 \( 1 + (2.71 + 3.95i)T \)
good7 \( 1 + 2.14T + 7T^{2} \)
11 \( 1 - 3.55T + 11T^{2} \)
13 \( 1 + 4.04T + 13T^{2} \)
17 \( 1 - 0.843iT - 17T^{2} \)
19 \( 1 - 5.13T + 19T^{2} \)
29 \( 1 - 0.275T + 29T^{2} \)
31 \( 1 - 8.41iT - 31T^{2} \)
37 \( 1 + 2.63iT - 37T^{2} \)
41 \( 1 - 7.36T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 - 10.4iT - 47T^{2} \)
53 \( 1 + 5.60iT - 53T^{2} \)
59 \( 1 - 2.95iT - 59T^{2} \)
61 \( 1 - 7.05iT - 61T^{2} \)
67 \( 1 - 2.62T + 67T^{2} \)
71 \( 1 - 2.73iT - 71T^{2} \)
73 \( 1 - 4.84T + 73T^{2} \)
79 \( 1 - 16.2T + 79T^{2} \)
83 \( 1 - 9.45T + 83T^{2} \)
89 \( 1 - 16.0iT - 89T^{2} \)
97 \( 1 + 0.845iT - 97T^{2} \)
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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

−9.482340767351374189293967786331, −9.023760435097574152249721232657, −7.86199879012592720440150538923, −7.27055111194502731870570210874, −6.55712708690025863019776730704, −5.99877092245960514005091661433, −4.91443444616926306088343774276, −3.75087026254026393646697126945, −2.46550511251599798261645470430, −0.963613564762989586430856608353, 0.63946481727834715963573182137, 2.17898694479819774071854268137, 3.31362371365131297476975055293, 4.05194172631961771573770531688, 4.96082127260691907965753749795, 6.00737109400152262389918406401, 7.25873476591083400029583440042, 7.940130744565763980898085031728, 9.146264789969786335321999259386, 9.518717654273791413767605961132

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