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Properties

Label 2-714-17.13-c1-0-1
Degree $2$
Conductor $714$
Sign $0.196 - 0.980i$
Analytic cond. $5.70131$
Root an. cond. $2.38774$
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 + (−0.707 − 0.707i)3-s − 4-s + (−1.14 − 1.14i)5-s + (−0.707 + 0.707i)6-s + (0.707 − 0.707i)7-s + i·8-s + 1.00i·9-s + (−1.14 + 1.14i)10-s + (−1.66 + 1.66i)11-s + (0.707 + 0.707i)12-s − 6.05·13-s + (−0.707 − 0.707i)14-s + 1.61i·15-s + 16-s + (1.84 + 3.68i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.510 − 0.510i)5-s + (−0.288 + 0.288i)6-s + (0.267 − 0.267i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.361 + 0.361i)10-s + (−0.502 + 0.502i)11-s + (0.204 + 0.204i)12-s − 1.67·13-s + (−0.188 − 0.188i)14-s + 0.416i·15-s + 0.250·16-s + (0.448 + 0.893i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.196 - 0.980i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ 0.196 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132056 + 0.108202i\)
\(L(\frac12)\) \(\approx\) \(0.132056 + 0.108202i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-1.84 - 3.68i)T \)
good5 \( 1 + (1.14 + 1.14i)T + 5iT^{2} \)
11 \( 1 + (1.66 - 1.66i)T - 11iT^{2} \)
13 \( 1 + 6.05T + 13T^{2} \)
19 \( 1 + 0.520iT - 19T^{2} \)
23 \( 1 + (2.41 - 2.41i)T - 23iT^{2} \)
29 \( 1 + (-4.08 - 4.08i)T + 29iT^{2} \)
31 \( 1 + (5.46 + 5.46i)T + 31iT^{2} \)
37 \( 1 + (-1.86 - 1.86i)T + 37iT^{2} \)
41 \( 1 + (6.25 - 6.25i)T - 41iT^{2} \)
43 \( 1 - 13.0iT - 43T^{2} \)
47 \( 1 - 8.65T + 47T^{2} \)
53 \( 1 + 1.01iT - 53T^{2} \)
59 \( 1 - 13.5iT - 59T^{2} \)
61 \( 1 + (3.09 - 3.09i)T - 61iT^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + (1.81 + 1.81i)T + 71iT^{2} \)
73 \( 1 + (11.1 + 11.1i)T + 73iT^{2} \)
79 \( 1 + (-11.5 + 11.5i)T - 79iT^{2} \)
83 \( 1 + 16.2iT - 83T^{2} \)
89 \( 1 + 6.71T + 89T^{2} \)
97 \( 1 + (8.06 + 8.06i)T + 97iT^{2} \)
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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

−10.52744933273883785688503927147, −10.03548400643535700260273573467, −8.976767875279402271687580395065, −7.82199463687279718005488665223, −7.48662480253871459089251783932, −6.03734441108624901455304165168, −4.89802017110604780328916346472, −4.33287129572267562718924562450, −2.81321895824986133695254281293, −1.53498694030623481555451176121, 0.093216597108798051204243524235, 2.62798672597838876456547947362, 3.86529404683602576637717991379, 5.05370724717669292066042095475, 5.54796680711068035269446335179, 6.89917034076017735925788906176, 7.44281006903111061648023166214, 8.389269509648421538764281990218, 9.383245425315764844526669516602, 10.21226707230131142277551639498

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