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Properties

Label 2-850-17.13-c1-0-12
Degree $2$
Conductor $850$
Sign $0.995 + 0.0995i$
Analytic cond. $6.78728$
Root an. cond. $2.60524$
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  − i·2-s + (1.20 + 1.20i)3-s − 4-s + (1.20 − 1.20i)6-s + (−1.69 + 1.69i)7-s + i·8-s − 0.113i·9-s + (3.82 − 3.82i)11-s + (−1.20 − 1.20i)12-s + 2.28·13-s + (1.69 + 1.69i)14-s + 16-s + (2.20 + 3.48i)17-s − 0.113·18-s + 3.51i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.693 + 0.693i)3-s − 0.5·4-s + (0.490 − 0.490i)6-s + (−0.642 + 0.642i)7-s + 0.353i·8-s − 0.0377i·9-s + (1.15 − 1.15i)11-s + (−0.346 − 0.346i)12-s + 0.633·13-s + (0.454 + 0.454i)14-s + 0.250·16-s + (0.533 + 0.845i)17-s − 0.0266·18-s + 0.806i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $0.995 + 0.0995i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ 0.995 + 0.0995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91927 - 0.0957359i\)
\(L(\frac12)\) \(\approx\) \(1.91927 - 0.0957359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
17 \( 1 + (-2.20 - 3.48i)T \)
good3 \( 1 + (-1.20 - 1.20i)T + 3iT^{2} \)
7 \( 1 + (1.69 - 1.69i)T - 7iT^{2} \)
11 \( 1 + (-3.82 + 3.82i)T - 11iT^{2} \)
13 \( 1 - 2.28T + 13T^{2} \)
19 \( 1 - 3.51iT - 19T^{2} \)
23 \( 1 + (-0.585 + 0.585i)T - 23iT^{2} \)
29 \( 1 + (-0.384 - 0.384i)T + 29iT^{2} \)
31 \( 1 + (-6.72 - 6.72i)T + 31iT^{2} \)
37 \( 1 + (-3.12 - 3.12i)T + 37iT^{2} \)
41 \( 1 + (-4.39 + 4.39i)T - 41iT^{2} \)
43 \( 1 - 1.59iT - 43T^{2} \)
47 \( 1 + 4.11T + 47T^{2} \)
53 \( 1 - 1.11iT - 53T^{2} \)
59 \( 1 + 11.1iT - 59T^{2} \)
61 \( 1 + (9.01 - 9.01i)T - 61iT^{2} \)
67 \( 1 + 8.46T + 67T^{2} \)
71 \( 1 + (-9.13 - 9.13i)T + 71iT^{2} \)
73 \( 1 + (8.74 + 8.74i)T + 73iT^{2} \)
79 \( 1 + (-11.6 + 11.6i)T - 79iT^{2} \)
83 \( 1 + 10.9iT - 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + (10.5 + 10.5i)T + 97iT^{2} \)
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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

−10.11011589041733949187157819213, −9.304235116491410210669449440573, −8.745813118444673728117930096966, −8.150521327193829968748559488597, −6.40417464245121533167662348195, −5.88851736533733846938862430734, −4.37735831913318553723439499059, −3.48131699421275204466058173848, −3.01626556913000345363947524571, −1.29315785051910579237873072737, 1.12179841070356624199076886640, 2.64324677889858944767699601545, 3.89474997412572094335348671827, 4.80423137500355417876938162995, 6.20612262731479656590180427315, 6.93777470777950384812243056026, 7.48119545508334294088431474922, 8.317432252762831535670274140662, 9.376049635021317496680277709407, 9.744397230715259780447087357803

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