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Properties

Label 2-770-55.43-c1-0-18
Degree $2$
Conductor $770$
Sign $0.696 - 0.717i$
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 + (−0.888 − 0.888i)3-s + 1.00i·4-s + (1.38 + 1.75i)5-s − 1.25i·6-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s − 1.42i·9-s + (−0.257 + 2.22i)10-s + (2.56 − 2.09i)11-s + (0.888 − 0.888i)12-s + (1.35 − 1.35i)13-s + 1.00i·14-s + (0.323 − 2.79i)15-s − 1.00·16-s + (2.87 + 2.87i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.512 − 0.512i)3-s + 0.500i·4-s + (0.620 + 0.783i)5-s − 0.512i·6-s + (0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.474i·9-s + (−0.0815 + 0.702i)10-s + (0.774 − 0.632i)11-s + (0.256 − 0.256i)12-s + (0.375 − 0.375i)13-s + 0.267i·14-s + (0.0836 − 0.720i)15-s − 0.250·16-s + (0.698 + 0.698i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.696 - 0.717i$
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: \((2,\ 770,\ (\ :1/2),\ 0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87175 + 0.792051i\)
\(L(\frac12)\) \(\approx\) \(1.87175 + 0.792051i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 - 1.75i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-2.56 + 2.09i)T \)
good3 \( 1 + (0.888 + 0.888i)T + 3iT^{2} \)
13 \( 1 + (-1.35 + 1.35i)T - 13iT^{2} \)
17 \( 1 + (-2.87 - 2.87i)T + 17iT^{2} \)
19 \( 1 + 1.47T + 19T^{2} \)
23 \( 1 + (-4.72 - 4.72i)T + 23iT^{2} \)
29 \( 1 + 8.86T + 29T^{2} \)
31 \( 1 - 8.87T + 31T^{2} \)
37 \( 1 + (-1.13 + 1.13i)T - 37iT^{2} \)
41 \( 1 - 8.24iT - 41T^{2} \)
43 \( 1 + (-4.20 + 4.20i)T - 43iT^{2} \)
47 \( 1 + (-8.69 + 8.69i)T - 47iT^{2} \)
53 \( 1 + (0.533 + 0.533i)T + 53iT^{2} \)
59 \( 1 - 9.93iT - 59T^{2} \)
61 \( 1 + 2.09iT - 61T^{2} \)
67 \( 1 + (-7.31 + 7.31i)T - 67iT^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + (10.4 - 10.4i)T - 73iT^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + (-7.94 + 7.94i)T - 83iT^{2} \)
89 \( 1 + 8.22iT - 89T^{2} \)
97 \( 1 + (-1.76 + 1.76i)T - 97iT^{2} \)
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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

−10.60636694185790349648963297473, −9.489654910939719480124732663468, −8.656188336823314452730902420181, −7.52424586657988099096196580029, −6.76984277350377175329956950848, −5.88681875334215213443170583112, −5.62384091271431869235770413260, −3.94167668530964566088258109879, −3.02252084520552979683986686602, −1.40295580099852142623558351316, 1.16249654838679785112518865920, 2.42053968219246809879236823559, 4.10497907959510896804144200716, 4.68315592225775994282125884295, 5.49063282066468937266594634414, 6.39867973053874091630114100558, 7.57694133732782770145695747878, 8.840520891048093176331189675746, 9.530476087805228115632235709287, 10.30454495997381646091374533030

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