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Properties

Label 2-780-195.83-c0-0-1
Degree $2$
Conductor $780$
Sign $0.966 + 0.256i$
Analytic cond. $0.389270$
Root an. cond. $0.623915$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s + 7-s − 1.00i·9-s + (−0.707 + 0.707i)11-s − 13-s + 1.00·15-s + (−0.707 + 0.707i)17-s + (0.707 − 0.707i)21-s + (−0.707 − 0.707i)23-s + 1.00i·25-s + (−0.707 − 0.707i)27-s − 1.41·29-s + (1 − i)31-s + 1.00i·33-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s + 7-s − 1.00i·9-s + (−0.707 + 0.707i)11-s − 13-s + 1.00·15-s + (−0.707 + 0.707i)17-s + (0.707 − 0.707i)21-s + (−0.707 − 0.707i)23-s + 1.00i·25-s + (−0.707 − 0.707i)27-s − 1.41·29-s + (1 − i)31-s + 1.00i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.966 + 0.256i$
Analytic conductor: \(0.389270\)
Root analytic conductor: \(0.623915\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :0),\ 0.966 + 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.336135838\)
\(L(\frac12)\) \(\approx\) \(1.336135838\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + T \)
good7 \( 1 - T + T^{2} \)
11 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
17 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + (-1 + i)T - iT^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−10.34344886163109503064682379237, −9.674827134823979536303051257747, −8.658724999010855754258454345916, −7.78113779872982913787050690890, −7.19442947925607431341428353887, −6.24673308088798553496095624675, −5.17307789491501927818164143657, −3.96855625299123681893200329407, −2.38867478010405634551590184135, −2.02551986146899686750955544320, 1.87372940005070565694114541896, 2.88356162613494154554633034088, 4.39370048353639550006892322179, 5.02560723417937718690261809330, 5.80410351093157961574027623442, 7.44714941444174325819122147512, 8.085101980025009619956645776904, 8.988843737431236406135767376007, 9.513241373644548206312045831029, 10.47379480176644816793406296012

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