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Properties

Label 2-650-25.21-c1-0-13
Degree $2$
Conductor $650$
Sign $0.607 + 0.794i$
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  + (−0.309 + 0.951i)2-s + (−0.126 − 0.0917i)3-s + (−0.809 − 0.587i)4-s + (−1.23 + 1.86i)5-s + (0.126 − 0.0917i)6-s − 2.41·7-s + (0.809 − 0.587i)8-s + (−0.919 − 2.83i)9-s + (−1.39 − 1.74i)10-s + (−0.733 + 2.25i)11-s + (0.0482 + 0.148i)12-s + (−0.309 − 0.951i)13-s + (0.747 − 2.30i)14-s + (0.326 − 0.122i)15-s + (0.309 + 0.951i)16-s + (4.87 − 3.54i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.0728 − 0.0529i)3-s + (−0.404 − 0.293i)4-s + (−0.551 + 0.833i)5-s + (0.0515 − 0.0374i)6-s − 0.914·7-s + (0.286 − 0.207i)8-s + (−0.306 − 0.943i)9-s + (−0.440 − 0.553i)10-s + (−0.221 + 0.680i)11-s + (0.0139 + 0.0428i)12-s + (−0.0857 − 0.263i)13-s + (0.199 − 0.614i)14-s + (0.0843 − 0.0315i)15-s + (0.0772 + 0.237i)16-s + (1.18 − 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.607 + 0.794i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.607 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.524911 - 0.259439i\)
\(L(\frac12)\) \(\approx\) \(0.524911 - 0.259439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (1.23 - 1.86i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (0.126 + 0.0917i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 2.41T + 7T^{2} \)
11 \( 1 + (0.733 - 2.25i)T + (-8.89 - 6.46i)T^{2} \)
17 \( 1 + (-4.87 + 3.54i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.61 + 1.17i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.62 + 8.08i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.670 - 0.486i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (7.09 - 5.15i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.19 + 3.66i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.20 + 6.78i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 + (4.58 + 3.32i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (5.91 + 4.29i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.383 - 1.17i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.27 + 3.91i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (2.06 - 1.49i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (5.50 + 4.00i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.613 - 1.88i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.13 - 5.18i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.58 + 4.78i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.650 + 2.00i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (10.7 + 7.77i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.26117637878456632502344539752, −9.533958669318932747459062446546, −8.702106821830962555954800910341, −7.46507188014734972757837760917, −6.99086989819347481350419937743, −6.18233667125640657547976365631, −5.09597023776910059268099252596, −3.70453804185645627281125363715, −2.84160598927255381882949858309, −0.36792676686944837507364858791, 1.38107114072406274861837328251, 3.06492051627789575366017243539, 3.88604686805847829570970358424, 5.17056738922424776207964746051, 5.92067566326939009496847998892, 7.60549602290586884695644806787, 8.032526610606714870170725592701, 9.158942638859230275144792197195, 9.725936095209995023132323693832, 10.78669637673034567890850071216

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