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Properties

Label 2-770-11.5-c1-0-0
Degree $2$
Conductor $770$
Sign $-0.904 + 0.426i$
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 + (0.200 + 0.145i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.0763 + 0.235i)6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.908 − 2.79i)9-s − 0.999·10-s + (−3.26 + 0.569i)11-s − 0.247·12-s + (0.875 + 2.69i)13-s + (−0.809 − 0.587i)14-s + (−0.200 + 0.145i)15-s + (0.309 − 0.951i)16-s + (−1.78 + 5.48i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.115 + 0.0839i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (−0.0311 + 0.0959i)6-s + (−0.305 + 0.222i)7-s + (−0.286 − 0.207i)8-s + (−0.302 − 0.931i)9-s − 0.316·10-s + (−0.985 + 0.171i)11-s − 0.0713·12-s + (0.242 + 0.747i)13-s + (−0.216 − 0.157i)14-s + (−0.0516 + 0.0375i)15-s + (0.0772 − 0.237i)16-s + (−0.432 + 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.904 + 0.426i$
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: \((2,\ 770,\ (\ :1/2),\ -0.904 + 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116501 - 0.519750i\)
\(L(\frac12)\) \(\approx\) \(0.116501 - 0.519750i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.26 - 0.569i)T \)
good3 \( 1 + (-0.200 - 0.145i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (-0.875 - 2.69i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.78 - 5.48i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.96 + 2.15i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 7.17T + 23T^{2} \)
29 \( 1 + (1.14 - 0.831i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.110 + 0.338i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.38 - 1.73i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.80 + 1.30i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.64T + 43T^{2} \)
47 \( 1 + (2.53 + 1.84i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.270 - 0.831i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.49 + 6.16i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.84 - 8.75i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + (3.86 - 11.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.82 + 4.23i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.89 - 5.84i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.952 + 2.93i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 5.75T + 89T^{2} \)
97 \( 1 + (-3.87 - 11.9i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.69293969427280667805739035332, −9.892663050658267871883468545497, −8.880038588112454541121009517112, −8.294047555713537460113081624683, −7.22626454484125705116853540131, −6.34556543337892818602292001616, −5.79565358235914841363550255900, −4.38062285533514245174367682775, −3.60235244295807996970406918412, −2.29662959569247821481038481811, 0.22835835428174503491975571932, 2.08562292035430839655702261022, 3.05338124584267381810149059168, 4.29448776949474979995698404169, 5.21376742125387763032436580883, 6.01436038520787836283349881118, 7.51704244732364592211854021826, 8.128196772522126992036204632419, 9.030284448248333782705854598052, 10.08881959431659633149960573200

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