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Properties

Label 2-770-55.43-c1-0-21
Degree $2$
Conductor $770$
Sign $0.786 + 0.617i$
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 + (1.47 + 1.47i)3-s + 1.00i·4-s + (−2.23 − 0.0355i)5-s − 2.08i·6-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + 1.35i·9-s + (1.55 + 1.60i)10-s + (0.318 − 3.30i)11-s + (−1.47 + 1.47i)12-s + (2.95 − 2.95i)13-s + 1.00i·14-s + (−3.24 − 3.35i)15-s − 1.00·16-s + (0.451 + 0.451i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.852 + 0.852i)3-s + 0.500i·4-s + (−0.999 − 0.0158i)5-s − 0.852i·6-s + (−0.267 − 0.267i)7-s + (0.250 − 0.250i)8-s + 0.451i·9-s + (0.491 + 0.507i)10-s + (0.0959 − 0.995i)11-s + (−0.426 + 0.426i)12-s + (0.819 − 0.819i)13-s + 0.267i·14-s + (−0.838 − 0.865i)15-s − 0.250·16-s + (0.109 + 0.109i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.786 + 0.617i$
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: \((2,\ 770,\ (\ :1/2),\ 0.786 + 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23268 - 0.426081i\)
\(L(\frac12)\) \(\approx\) \(1.23268 - 0.426081i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.0355i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (-0.318 + 3.30i)T \)
good3 \( 1 + (-1.47 - 1.47i)T + 3iT^{2} \)
13 \( 1 + (-2.95 + 2.95i)T - 13iT^{2} \)
17 \( 1 + (-0.451 - 0.451i)T + 17iT^{2} \)
19 \( 1 - 4.23T + 19T^{2} \)
23 \( 1 + (-1.62 - 1.62i)T + 23iT^{2} \)
29 \( 1 + 2.38T + 29T^{2} \)
31 \( 1 - 3.62T + 31T^{2} \)
37 \( 1 + (0.785 - 0.785i)T - 37iT^{2} \)
41 \( 1 + 5.52iT - 41T^{2} \)
43 \( 1 + (-0.646 + 0.646i)T - 43iT^{2} \)
47 \( 1 + (0.355 - 0.355i)T - 47iT^{2} \)
53 \( 1 + (-7.85 - 7.85i)T + 53iT^{2} \)
59 \( 1 + 3.53iT - 59T^{2} \)
61 \( 1 - 3.09iT - 61T^{2} \)
67 \( 1 + (-11.1 + 11.1i)T - 67iT^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + (1.42 - 1.42i)T - 73iT^{2} \)
79 \( 1 + 2.62T + 79T^{2} \)
83 \( 1 + (5.82 - 5.82i)T - 83iT^{2} \)
89 \( 1 + 2.20iT - 89T^{2} \)
97 \( 1 + (-10.4 + 10.4i)T - 97iT^{2} \)
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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

−10.23039004776251345167606752752, −9.311402694846673064282539451805, −8.597937961007792747208828832088, −8.050536241985086628486876800295, −7.09551149723077973126295253641, −5.69257119734942672389977845272, −4.28981577761887989430884861292, −3.43626466039668495720517006811, −3.04709270426178710762362078994, −0.840032684242266731233130232942, 1.30295475251030261653503793856, 2.62599631255628494244276902534, 3.88016651649944072985741680689, 5.05990681809872527709371563401, 6.51189541214545063476945824428, 7.14764063319941252492494980074, 7.79251365968714887566794212157, 8.576525184550491938116106306091, 9.202606298300233953433509312380, 10.19237315062183893697986602517

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