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Properties

Label 2-770-11.5-c1-0-18
Degree $2$
Conductor $770$
Sign $-0.826 + 0.562i$
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  + (0.309 + 0.951i)2-s + (−1.97 − 1.43i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (0.755 − 2.32i)6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.920 + 2.83i)9-s + 0.999·10-s + (3.13 − 1.07i)11-s + 2.44·12-s + (0.310 + 0.955i)13-s + (−0.809 − 0.587i)14-s + (−1.97 + 1.43i)15-s + (0.309 − 0.951i)16-s + (0.301 − 0.926i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−1.14 − 0.829i)3-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (0.308 − 0.949i)6-s + (−0.305 + 0.222i)7-s + (−0.286 − 0.207i)8-s + (0.306 + 0.944i)9-s + 0.316·10-s + (0.946 − 0.323i)11-s + 0.705·12-s + (0.0860 + 0.264i)13-s + (−0.216 − 0.157i)14-s + (−0.510 + 0.371i)15-s + (0.0772 − 0.237i)16-s + (0.0730 − 0.224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.826 + 0.562i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.826 + 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0941615 - 0.305841i\)
\(L(\frac12)\) \(\approx\) \(0.0941615 - 0.305841i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-3.13 + 1.07i)T \)
good3 \( 1 + (1.97 + 1.43i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (-0.310 - 0.955i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.301 + 0.926i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.15 + 3.01i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.97T + 23T^{2} \)
29 \( 1 + (2.34 - 1.70i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.99 + 6.15i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.92 - 5.75i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.50 + 1.82i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.46T + 43T^{2} \)
47 \( 1 + (5.90 + 4.28i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.23 - 3.79i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.729 + 0.530i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.19 + 6.76i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.88T + 67T^{2} \)
71 \( 1 + (-0.682 + 2.10i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.57 + 2.59i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.94 - 12.1i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.76 + 8.51i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 5.28T + 89T^{2} \)
97 \( 1 + (2.90 + 8.95i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.879539099752671412812089734090, −8.956774127732916688090988198975, −8.158041534847591219264665320080, −6.93302770604095012306481897543, −6.49871349315970597327939580029, −5.73430250664443792916830907663, −4.86679213151737069163410683927, −3.68292129076765261166366422251, −1.76698259139970037818572488649, −0.17162509590706195118572070361, 1.79583689561980103589442118572, 3.52428235631741518880659179969, 4.16835501427329493432564594092, 5.23569515234690739077331983858, 6.09733390671107077427774693593, 6.82412171752676594662486653469, 8.289679359835287210416204574856, 9.404222072627866215033857612862, 10.20192508091582977359952044569, 10.53360650757946747317910792826

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