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Properties

Label 2-1700-17.4-c1-0-21
Degree $2$
Conductor $1700$
Sign $0.509 + 0.860i$
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  + (−0.254 + 0.254i)3-s + (0.964 + 0.964i)7-s + 2.87i·9-s + (−2.68 − 2.68i)11-s − 4.50·13-s + (0.529 − 4.08i)17-s − 7.03i·19-s − 0.491·21-s + (1.68 + 1.68i)23-s + (−1.49 − 1.49i)27-s + (5.95 − 5.95i)29-s + (3.75 − 3.75i)31-s + 1.36·33-s + (5.50 − 5.50i)37-s + (1.14 − 1.14i)39-s + ⋯
L(s)  = 1  + (−0.146 + 0.146i)3-s + (0.364 + 0.364i)7-s + 0.956i·9-s + (−0.810 − 0.810i)11-s − 1.24·13-s + (0.128 − 0.991i)17-s − 1.61i·19-s − 0.107·21-s + (0.352 + 0.352i)23-s + (−0.287 − 0.287i)27-s + (1.10 − 1.10i)29-s + (0.674 − 0.674i)31-s + 0.238·33-s + (0.905 − 0.905i)37-s + (0.183 − 0.183i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.509 + 0.860i$
Analytic conductor: \(13.5745\)
Root analytic conductor: \(3.68436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :1/2),\ 0.509 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.236565559\)
\(L(\frac12)\) \(\approx\) \(1.236565559\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.529 + 4.08i)T \)
good3 \( 1 + (0.254 - 0.254i)T - 3iT^{2} \)
7 \( 1 + (-0.964 - 0.964i)T + 7iT^{2} \)
11 \( 1 + (2.68 + 2.68i)T + 11iT^{2} \)
13 \( 1 + 4.50T + 13T^{2} \)
19 \( 1 + 7.03iT - 19T^{2} \)
23 \( 1 + (-1.68 - 1.68i)T + 23iT^{2} \)
29 \( 1 + (-5.95 + 5.95i)T - 29iT^{2} \)
31 \( 1 + (-3.75 + 3.75i)T - 31iT^{2} \)
37 \( 1 + (-5.50 + 5.50i)T - 37iT^{2} \)
41 \( 1 + (-7.39 - 7.39i)T + 41iT^{2} \)
43 \( 1 - 7.49iT - 43T^{2} \)
47 \( 1 + 8.72T + 47T^{2} \)
53 \( 1 - 5.05iT - 53T^{2} \)
59 \( 1 + 5.18iT - 59T^{2} \)
61 \( 1 + (4.05 + 4.05i)T + 61iT^{2} \)
67 \( 1 - 15.9T + 67T^{2} \)
71 \( 1 + (-9.96 + 9.96i)T - 71iT^{2} \)
73 \( 1 + (9.97 - 9.97i)T - 73iT^{2} \)
79 \( 1 + (-4.30 - 4.30i)T + 79iT^{2} \)
83 \( 1 + 15.6iT - 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + (7.17 - 7.17i)T - 97iT^{2} \)
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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

−9.438654535272957989910423963477, −8.143468909604884010199685356172, −7.86046816651322012724727727305, −6.87671463977039211113268487929, −5.82191518770075094183657790716, −4.90334350979376127460743828170, −4.64193727062648422338942990303, −2.80406862646868054496946799017, −2.45435997032388103406844579034, −0.51929028132355137298422545202, 1.19836908610273058602136106599, 2.42390792509043979569028591055, 3.58789548882708279209281225805, 4.55462930406674937081104173080, 5.33558781529022269353134479730, 6.35166139891272835741153067395, 7.09238500864757624769684757583, 7.86783058126695910859624745724, 8.549183497797820212124805559480, 9.666115852092278271841562192533

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