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Properties

Label 2-770-55.43-c1-0-27
Degree $2$
Conductor $770$
Sign $0.757 + 0.652i$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
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  + (−0.707 − 0.707i)2-s + (0.618 + 0.618i)3-s + 1.00i·4-s + (2 + i)5-s − 0.874i·6-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s − 2.23i·9-s + (−0.707 − 2.12i)10-s + (−1 − 3.16i)11-s + (−0.618 + 0.618i)12-s + (4.57 − 4.57i)13-s − 1.00i·14-s + (0.618 + 1.85i)15-s − 1.00·16-s + (−3.70 − 3.70i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.356 + 0.356i)3-s + 0.500i·4-s + (0.894 + 0.447i)5-s − 0.356i·6-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s − 0.745i·9-s + (−0.223 − 0.670i)10-s + (−0.301 − 0.953i)11-s + (−0.178 + 0.178i)12-s + (1.26 − 1.26i)13-s − 0.267i·14-s + (0.159 + 0.478i)15-s − 0.250·16-s + (−0.897 − 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.757 + 0.652i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.757 + 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51472 - 0.562330i\)
\(L(\frac12)\) \(\approx\) \(1.51472 - 0.562330i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-2 - i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (1 + 3.16i)T \)
good3 \( 1 + (-0.618 - 0.618i)T + 3iT^{2} \)
13 \( 1 + (-4.57 + 4.57i)T - 13iT^{2} \)
17 \( 1 + (3.70 + 3.70i)T + 17iT^{2} \)
19 \( 1 + 3.36T + 19T^{2} \)
23 \( 1 + (-1.85 - 1.85i)T + 23iT^{2} \)
29 \( 1 - 5.78T + 29T^{2} \)
31 \( 1 - 5.70T + 31T^{2} \)
37 \( 1 + (-5.38 + 5.38i)T - 37iT^{2} \)
41 \( 1 - 10.7iT - 41T^{2} \)
43 \( 1 + (4.24 - 4.24i)T - 43iT^{2} \)
47 \( 1 + (8.70 - 8.70i)T - 47iT^{2} \)
53 \( 1 + (6.61 + 6.61i)T + 53iT^{2} \)
59 \( 1 - 2.76iT - 59T^{2} \)
61 \( 1 - 2.82iT - 61T^{2} \)
67 \( 1 + (1.76 - 1.76i)T - 67iT^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (7.73 - 7.73i)T - 73iT^{2} \)
79 \( 1 - 8.61T + 79T^{2} \)
83 \( 1 + (1.74 - 1.74i)T - 83iT^{2} \)
89 \( 1 + 17.4iT - 89T^{2} \)
97 \( 1 + (-2.32 + 2.32i)T - 97iT^{2} \)
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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

−10.13202340645548329549018158314, −9.462290059086459981065967207675, −8.603801040072410551659430309494, −8.093683907115812247429240136229, −6.55456875157984838286316315327, −6.00108957805155235559984049120, −4.67788783798891733434896698754, −3.23116172374562214692116917409, −2.73998974935498054872227175846, −1.04536672960511457949221853833, 1.54965311212279088476321487505, 2.23754584438027759947358721070, 4.31152480589164532733961860849, 5.02668082857552376740864075032, 6.44493502172859246870840857776, 6.72482519612291304317546987338, 8.143962338096737919690115044926, 8.532234121743952250494184751031, 9.364652322193944510040632746284, 10.41205276889824609837003845764

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