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Properties

Label 2-690-15.8-c1-0-2
Degree $2$
Conductor $690$
Sign $-0.890 - 0.454i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  + (0.707 + 0.707i)2-s + (−1.07 − 1.35i)3-s + 1.00i·4-s + (1.85 + 1.24i)5-s + (0.194 − 1.72i)6-s + (−2.75 + 2.75i)7-s + (−0.707 + 0.707i)8-s + (−0.668 + 2.92i)9-s + (0.430 + 2.19i)10-s + 2.29i·11-s + (1.35 − 1.07i)12-s + (−4.98 − 4.98i)13-s − 3.90·14-s + (−0.314 − 3.86i)15-s − 1.00·16-s + (−3.92 − 3.92i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.623 − 0.781i)3-s + 0.500i·4-s + (0.829 + 0.557i)5-s + (0.0792 − 0.702i)6-s + (−1.04 + 1.04i)7-s + (−0.250 + 0.250i)8-s + (−0.222 + 0.974i)9-s + (0.136 + 0.693i)10-s + 0.692i·11-s + (0.390 − 0.311i)12-s + (−1.38 − 1.38i)13-s − 1.04·14-s + (−0.0812 − 0.996i)15-s − 0.250·16-s + (−0.950 − 0.950i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.890 - 0.454i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.890 - 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197329 + 0.820069i\)
\(L(\frac12)\) \(\approx\) \(0.197329 + 0.820069i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (1.07 + 1.35i)T \)
5 \( 1 + (-1.85 - 1.24i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (2.75 - 2.75i)T - 7iT^{2} \)
11 \( 1 - 2.29iT - 11T^{2} \)
13 \( 1 + (4.98 + 4.98i)T + 13iT^{2} \)
17 \( 1 + (3.92 + 3.92i)T + 17iT^{2} \)
19 \( 1 - 4.71iT - 19T^{2} \)
29 \( 1 + 6.12T + 29T^{2} \)
31 \( 1 - 0.118T + 31T^{2} \)
37 \( 1 + (-0.206 + 0.206i)T - 37iT^{2} \)
41 \( 1 - 5.80iT - 41T^{2} \)
43 \( 1 + (-5.81 - 5.81i)T + 43iT^{2} \)
47 \( 1 + (-7.73 - 7.73i)T + 47iT^{2} \)
53 \( 1 + (3.05 - 3.05i)T - 53iT^{2} \)
59 \( 1 + 9.92T + 59T^{2} \)
61 \( 1 - 9.71T + 61T^{2} \)
67 \( 1 + (0.772 - 0.772i)T - 67iT^{2} \)
71 \( 1 + 6.32iT - 71T^{2} \)
73 \( 1 + (0.302 + 0.302i)T + 73iT^{2} \)
79 \( 1 - 14.8iT - 79T^{2} \)
83 \( 1 + (-9.99 + 9.99i)T - 83iT^{2} \)
89 \( 1 - 5.52T + 89T^{2} \)
97 \( 1 + (-1.89 + 1.89i)T - 97iT^{2} \)
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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

−10.90478572927508207678698599299, −9.873181827375649953020243746689, −9.272102496863031422863900968664, −7.78406398439358552790247982807, −7.18570239403008414868284434191, −6.20724401100308162053068701018, −5.72602850945112293966522458139, −4.87011707480430751764130892254, −2.93523225530207050109965727274, −2.27046388947891511562747392318, 0.37103849265961288983721000247, 2.25743640919921125764631937715, 3.76881431292324181672142652676, 4.44227978017287807774564243422, 5.40198465365860223282224788786, 6.39637109150902806483356652075, 7.00488049530481945704932824418, 9.046352592372781224231994867717, 9.317905754899365849382919405358, 10.28445242093246416645357461615

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