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Properties

Label 2-775-155.123-c1-0-12
Degree $2$
Conductor $775$
Sign $0.742 - 0.669i$
Analytic cond. $6.18840$
Root an. cond. $2.48765$
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  + (1.73 − 1.73i)2-s + (−1.58 + 1.58i)3-s − 3.99i·4-s + 5.47i·6-s + (−1.73 + 1.73i)7-s + (−3.46 − 3.46i)8-s − 2.00i·9-s + 5.47i·11-s + (6.32 + 6.32i)12-s + 5.99i·14-s − 3.99·16-s + (−1.58 − 1.58i)17-s + (−3.46 − 3.46i)18-s + 7i·19-s − 5.47i·21-s + (9.48 + 9.48i)22-s + ⋯
L(s)  = 1  + (1.22 − 1.22i)2-s + (−0.912 + 0.912i)3-s − 1.99i·4-s + 2.23i·6-s + (−0.654 + 0.654i)7-s + (−1.22 − 1.22i)8-s − 0.666i·9-s + 1.65i·11-s + (1.82 + 1.82i)12-s + 1.60i·14-s − 0.999·16-s + (−0.383 − 0.383i)17-s + (−0.816 − 0.816i)18-s + 1.60i·19-s − 1.19i·21-s + (2.02 + 2.02i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.742 - 0.669i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.742 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44763 + 0.556625i\)
\(L(\frac12)\) \(\approx\) \(1.44763 + 0.556625i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 + (1 + 5.47i)T \)
good2 \( 1 + (-1.73 + 1.73i)T - 2iT^{2} \)
3 \( 1 + (1.58 - 1.58i)T - 3iT^{2} \)
7 \( 1 + (1.73 - 1.73i)T - 7iT^{2} \)
11 \( 1 - 5.47iT - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (1.58 + 1.58i)T + 17iT^{2} \)
19 \( 1 - 7iT - 19T^{2} \)
23 \( 1 + (3.16 - 3.16i)T - 23iT^{2} \)
29 \( 1 - 5.47T + 29T^{2} \)
37 \( 1 + (-4.74 - 4.74i)T + 37iT^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + (4.74 - 4.74i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (7.90 - 7.90i)T - 53iT^{2} \)
59 \( 1 - 3iT - 59T^{2} \)
61 \( 1 + 10.9iT - 61T^{2} \)
67 \( 1 + (8.66 - 8.66i)T - 67iT^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 + (-4.74 + 4.74i)T - 73iT^{2} \)
79 \( 1 + 5.47T + 79T^{2} \)
83 \( 1 + (-7.90 + 7.90i)T - 83iT^{2} \)
89 \( 1 + 5.47T + 89T^{2} \)
97 \( 1 + (-10.3 + 10.3i)T - 97iT^{2} \)
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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

−10.51707462382421037900829338727, −9.801465889225428925249779969875, −9.555996306211802619823113012830, −7.75020139813155433445951203931, −6.20644363786776097379486752226, −5.73755994435906141011821364297, −4.61677813364547738151777230325, −4.26722962749976135008584376533, −3.00338921324611947263027893202, −1.84340842523270487923656087311, 0.57904563875377422959986203556, 2.97054957013135475697376262720, 4.03450727455848983956399887030, 5.12631345943810026898060863394, 6.05061593204208247235299756200, 6.51304870881676611467263624808, 7.09323625274680231661248586110, 8.070702752330122699388514715302, 8.975334302101233327203242526036, 10.56371606987095170421231496028

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