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Properties

Label 2-1700-85.4-c1-0-0
Degree $2$
Conductor $1700$
Sign $-0.998 - 0.0573i$
Analytic cond. $13.5745$
Root an. cond. $3.68436$
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  + (0.0789 + 0.0789i)3-s + (−1.97 + 1.97i)7-s − 2.98i·9-s + (−0.864 − 0.864i)11-s + 3.40i·13-s + (−1.04 + 3.98i)17-s − 3.42i·19-s − 0.311·21-s + (6.74 − 6.74i)23-s + (0.472 − 0.472i)27-s + (−7.08 + 7.08i)29-s + (−5.64 + 5.64i)31-s − 0.136i·33-s + (−4.04 − 4.04i)37-s + (−0.268 + 0.268i)39-s + ⋯
L(s)  = 1  + (0.0456 + 0.0456i)3-s + (−0.745 + 0.745i)7-s − 0.995i·9-s + (−0.260 − 0.260i)11-s + 0.943i·13-s + (−0.253 + 0.967i)17-s − 0.785i·19-s − 0.0680·21-s + (1.40 − 1.40i)23-s + (0.0910 − 0.0910i)27-s + (−1.31 + 1.31i)29-s + (−1.01 + 1.01i)31-s − 0.0237i·33-s + (−0.664 − 0.664i)37-s + (−0.0430 + 0.0430i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.998 - 0.0573i$
Analytic conductor: \(13.5745\)
Root analytic conductor: \(3.68436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :1/2),\ -0.998 - 0.0573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1502577791\)
\(L(\frac12)\) \(\approx\) \(0.1502577791\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 + (1.04 - 3.98i)T \)
good3 \( 1 + (-0.0789 - 0.0789i)T + 3iT^{2} \)
7 \( 1 + (1.97 - 1.97i)T - 7iT^{2} \)
11 \( 1 + (0.864 + 0.864i)T + 11iT^{2} \)
13 \( 1 - 3.40iT - 13T^{2} \)
19 \( 1 + 3.42iT - 19T^{2} \)
23 \( 1 + (-6.74 + 6.74i)T - 23iT^{2} \)
29 \( 1 + (7.08 - 7.08i)T - 29iT^{2} \)
31 \( 1 + (5.64 - 5.64i)T - 31iT^{2} \)
37 \( 1 + (4.04 + 4.04i)T + 37iT^{2} \)
41 \( 1 + (8.23 + 8.23i)T + 41iT^{2} \)
43 \( 1 + 7.49T + 43T^{2} \)
47 \( 1 - 8.61iT - 47T^{2} \)
53 \( 1 + 3.58T + 53T^{2} \)
59 \( 1 + 4.23iT - 59T^{2} \)
61 \( 1 + (7.67 + 7.67i)T + 61iT^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 + (-0.266 + 0.266i)T - 71iT^{2} \)
73 \( 1 + (6.97 + 6.97i)T + 73iT^{2} \)
79 \( 1 + (-10.2 - 10.2i)T + 79iT^{2} \)
83 \( 1 - 1.52T + 83T^{2} \)
89 \( 1 + 4.28T + 89T^{2} \)
97 \( 1 + (1.80 + 1.80i)T + 97iT^{2} \)
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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

−9.358144588754804922119032217288, −9.049847191602532017720646771590, −8.487244794906515953948323359743, −6.92804452216416926442160278099, −6.75497701706470593076714071301, −5.73433542345901339514270905998, −4.85335439723911362922847780653, −3.67174916708746020091107326363, −2.99035873338696566335094540605, −1.71062435746829026212227385619, 0.05352653763209541056820793571, 1.69377992170512086110971176970, 2.97215240066922580130727967653, 3.73578022500008078046397260483, 4.96872613469380546337058737033, 5.53766824730589074208774951117, 6.68001935086234007638759278345, 7.54470356510860766250433619627, 7.85022320316425843208560875138, 9.056641719402263820712045282790

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