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Properties

Label 2-650-13.12-c1-0-2
Degree $2$
Conductor $650$
Sign $-0.935 - 0.352i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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 − 3.10·3-s − 4-s − 3.10i·6-s − 4.10i·7-s − i·8-s + 6.64·9-s + 1.53i·11-s + 3.10·12-s + (−1.26 + 3.37i)13-s + 4.10·14-s + 16-s + 3·17-s + 6.64i·18-s + 1.10i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.79·3-s − 0.5·4-s − 1.26i·6-s − 1.55i·7-s − 0.353i·8-s + 2.21·9-s + 0.463i·11-s + 0.896·12-s + (−0.352 + 0.935i)13-s + 1.09·14-s + 0.250·16-s + 0.727·17-s + 1.56i·18-s + 0.253i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.935 - 0.352i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.935 - 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0519697 + 0.285805i\)
\(L(\frac12)\) \(\approx\) \(0.0519697 + 0.285805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
13 \( 1 + (1.26 - 3.37i)T \)
good3 \( 1 + 3.10T + 3T^{2} \)
7 \( 1 + 4.10iT - 7T^{2} \)
11 \( 1 - 1.53iT - 11T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 1.10iT - 19T^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 + 5.56T + 29T^{2} \)
31 \( 1 + 8.64iT - 31T^{2} \)
37 \( 1 - 4.53iT - 37T^{2} \)
41 \( 1 - 7.85iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 10.8iT - 47T^{2} \)
53 \( 1 + 0.433T + 53T^{2} \)
59 \( 1 - 13.7iT - 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 - 3.74iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 + 6.53T + 79T^{2} \)
83 \( 1 - 4.46iT - 83T^{2} \)
89 \( 1 - 1.85iT - 89T^{2} \)
97 \( 1 - 3.78iT - 97T^{2} \)
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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

−10.94813139245981654525645139334, −10.02628139538757264968958390132, −9.632925048952766338410627618643, −7.74285711157645616856218635330, −7.32152738124582596236492201277, −6.38035929936936798423121604957, −5.73675751433122860082023157444, −4.47267322201668258080984084349, −4.14067081897080512563816941189, −1.28785765409939002045806210630, 0.21906021071654883319821472588, 1.93062192838150712137672572950, 3.44878849751279892031049119075, 4.93666724027625939723634345295, 5.58303482469784302203427372505, 6.08155006976769597520581385817, 7.45525235001863316773384925291, 8.614662053243494881541147347407, 9.621916422348969033957904797477, 10.44730222702377354652527360503

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