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Properties

Label 2-31e2-31.25-c1-0-25
Degree $2$
Conductor $961$
Sign $-0.198 - 0.980i$
Analytic cond. $7.67362$
Root an. cond. $2.77013$
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  + 1.23·2-s + (1.03 + 1.79i)3-s − 0.481·4-s + (0.772 − 1.33i)5-s + (1.27 + 2.21i)6-s + (1.90 + 3.29i)7-s − 3.05·8-s + (−0.652 + 1.13i)9-s + (0.952 − 1.64i)10-s + (−1.88 + 3.25i)11-s + (−0.499 − 0.865i)12-s + (−1.31 + 2.28i)13-s + (2.34 + 4.05i)14-s + 3.20·15-s − 2.80·16-s + (1.88 + 3.26i)17-s + ⋯
L(s)  = 1  + 0.871·2-s + (0.598 + 1.03i)3-s − 0.240·4-s + (0.345 − 0.598i)5-s + (0.521 + 0.903i)6-s + (0.718 + 1.24i)7-s − 1.08·8-s + (−0.217 + 0.376i)9-s + (0.301 − 0.521i)10-s + (−0.567 + 0.982i)11-s + (−0.144 − 0.249i)12-s + (−0.365 + 0.632i)13-s + (0.626 + 1.08i)14-s + 0.828·15-s − 0.701·16-s + (0.457 + 0.792i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66730 + 2.03874i\)
\(L(\frac12)\) \(\approx\) \(1.66730 + 2.03874i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 - 1.23T + 2T^{2} \)
3 \( 1 + (-1.03 - 1.79i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.772 + 1.33i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.90 - 3.29i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.88 - 3.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.31 - 2.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.88 - 3.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.04 + 5.28i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.909T + 23T^{2} \)
29 \( 1 - 6.80T + 29T^{2} \)
37 \( 1 + (0.907 + 1.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.168 - 0.291i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.94 - 3.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.18T + 47T^{2} \)
53 \( 1 + (1.17 - 2.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.88 + 6.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 2.72T + 61T^{2} \)
67 \( 1 + (-3.71 + 6.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.54 + 4.41i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.69 + 4.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.86 + 8.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.19 - 7.27i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.09T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−10.01133414451141176446494425745, −9.293221098028278890231631501984, −8.840744002761554716686526916663, −8.111862648128875358728419066753, −6.57409599245318858297361280937, −5.46144359443939190033942403675, −4.75195786227952216235580756954, −4.42423142343576583214474953089, −3.07530190049712705528009159976, −2.08822672563848085139268456632, 0.894467691838056026675557165916, 2.48471578950091099789633525741, 3.30716115819003463531170463043, 4.41923129164094750717567171218, 5.41928839828983641570789881217, 6.36810362254998016039554090767, 7.25131839422144406057580025806, 8.065850598562140721816245284961, 8.547500375774421246615738325231, 10.08819484511476657061010680957

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