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Properties

Label 2-690-15.14-c2-0-0
Degree $2$
Conductor $690$
Sign $-0.572 + 0.820i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + (−1.95 + 2.27i)3-s + 2.00·4-s + (−4.97 + 0.501i)5-s + (−2.76 + 3.21i)6-s + 8.64i·7-s + 2.82·8-s + (−1.35 − 8.89i)9-s + (−7.03 + 0.709i)10-s − 3.45i·11-s + (−3.90 + 4.55i)12-s + 8.53i·13-s + 12.2i·14-s + (8.58 − 12.3i)15-s + 4.00·16-s − 0.0194·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.651 + 0.758i)3-s + 0.500·4-s + (−0.994 + 0.100i)5-s + (−0.460 + 0.536i)6-s + 1.23i·7-s + 0.353·8-s + (−0.150 − 0.988i)9-s + (−0.703 + 0.0709i)10-s − 0.314i·11-s + (−0.325 + 0.379i)12-s + 0.656i·13-s + 0.873i·14-s + (0.572 − 0.820i)15-s + 0.250·16-s − 0.00114·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.820i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.572 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.572 + 0.820i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.572 + 0.820i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1045945386\)
\(L(\frac12)\) \(\approx\) \(0.1045945386\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (1.95 - 2.27i)T \)
5 \( 1 + (4.97 - 0.501i)T \)
23 \( 1 - 4.79T \)
good7 \( 1 - 8.64iT - 49T^{2} \)
11 \( 1 + 3.45iT - 121T^{2} \)
13 \( 1 - 8.53iT - 169T^{2} \)
17 \( 1 + 0.0194T + 289T^{2} \)
19 \( 1 + 36.9T + 361T^{2} \)
29 \( 1 + 17.7iT - 841T^{2} \)
31 \( 1 + 15.6T + 961T^{2} \)
37 \( 1 + 54.6iT - 1.36e3T^{2} \)
41 \( 1 + 31.5iT - 1.68e3T^{2} \)
43 \( 1 + 49.5iT - 1.84e3T^{2} \)
47 \( 1 + 69.5T + 2.20e3T^{2} \)
53 \( 1 + 45.7T + 2.80e3T^{2} \)
59 \( 1 + 32.1iT - 3.48e3T^{2} \)
61 \( 1 - 57.6T + 3.72e3T^{2} \)
67 \( 1 + 36.6iT - 4.48e3T^{2} \)
71 \( 1 - 100. iT - 5.04e3T^{2} \)
73 \( 1 - 130. iT - 5.32e3T^{2} \)
79 \( 1 + 142.T + 6.24e3T^{2} \)
83 \( 1 + 28.0T + 6.88e3T^{2} \)
89 \( 1 + 148. iT - 7.92e3T^{2} \)
97 \( 1 - 102. iT - 9.40e3T^{2} \)
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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

−11.22139422231130225545227207708, −10.21233649273441949215878186337, −8.985121197351888598571316284291, −8.435624040726235862948217565244, −7.02094695982488133723894165356, −6.18315838467355347790241194059, −5.34689886565941902511557670143, −4.35619202296691804322730417312, −3.65291563091358562785697891896, −2.32194815335697437145101986583, 0.03193558787334536274344907210, 1.42534406272871039315881939533, 3.08903446712828025201575514821, 4.30673858923074757790739530359, 4.90161358814379263131760556122, 6.30077048289604176285861742918, 6.92708980080481376196092697800, 7.75042618718144809960728481081, 8.390625941374651964327774582059, 10.14611487033595851698465794278

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