Properties

Label 2-614-307.289-c1-0-26
Degree $2$
Conductor $614$
Sign $-0.744 - 0.667i$
Analytic cond. $4.90281$
Root an. cond. $2.21423$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 4-s + (−2 − 3.46i)5-s − 3·6-s + 8-s + 6·9-s + (−2 − 3.46i)10-s + (−1.5 − 2.59i)11-s − 3·12-s + (3 + 5.19i)13-s + (6 + 10.3i)15-s + 16-s − 7·17-s + 6·18-s − 4·19-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 0.5·4-s + (−0.894 − 1.54i)5-s − 1.22·6-s + 0.353·8-s + 2·9-s + (−0.632 − 1.09i)10-s + (−0.452 − 0.783i)11-s − 0.866·12-s + (0.832 + 1.44i)13-s + (1.54 + 2.68i)15-s + 0.250·16-s − 1.69·17-s + 1.41·18-s − 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(614\)    =    \(2 \cdot 307\)
Sign: $-0.744 - 0.667i$
Analytic conductor: \(4.90281\)
Root analytic conductor: \(2.21423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{614} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 614,\ (\ :1/2),\ -0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
307 \( 1 + (8 + 15.5i)T \)
good3 \( 1 + 3T + 3T^{2} \)
5 \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.5 + 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (4 - 6.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−10.65196960892795150609847448475, −9.102737053950518779678900393431, −8.381265299917251239658185324954, −7.04482225332895130230973359205, −6.22055861452798134902564778165, −5.38865811648674766328963408122, −4.45661793875446283633316228410, −4.08333048207570170320155636318, −1.52358858231315068307710758304, 0, 2.47155195514422608380179301057, 3.88729562333261088718551048360, 4.68279331878965062889719623598, 5.89629396689026845443342249930, 6.52422749793746797491439416523, 7.17124696425949590876184801465, 8.147635050496111421369184233439, 10.14798684550299299211691660959, 10.70287213437937646080513526332

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