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Properties

Label 2-1700-17.13-c1-0-5
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: $\chi_{1700} (1101, \cdot )$
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

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

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